🕷️ Crawler Inspector

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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 152 (from laksa016)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa152.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Error_detection_and_correction\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Error_detection_and_correction\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction","crawl_time":1774109309,"first_indexed_time":1379787301,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Error_detection_and_correction s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Error detection and correction - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"To clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied\nReed–Solomon error correction\n(right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted.\nIn\ninformation theory\nand\ncoding theory\nwith applications in\ncomputer science\nand\ntelecommunications\n,\nerror detection and correction\n(\nEDAC\n) or\nerror control\nare techniques that enable\nreliable delivery\nof\ndigital data\nover unreliable\ncommunication channels\n. Many communication channels are subject to\nchannel noise\n, and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases.\nError detection\nis the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver.\nError correction\nis the detection of errors and reconstruction of the original, error-free data.\nIn classical antiquity,\ncopyists\nof the\nHebrew Bible\nwere paid for their work according to the number of\nstichs\n(lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters.\n[\n1\n]\nThis also helped ensure accuracy in the transmission of the text with the production of subsequent copies.\n[\n2\n]\n[\n3\n]\nBetween the 7th and 10th centuries CE a\ngroup of Jewish scribes\nformalized and expanded this to create the\nNumerical Masorah\nto ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary.\n[\n1\n]\nStandards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable.\n[\n4\n]\nThe effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the\nDead Sea Scrolls\nin 1947–1956, dating from\nc.\n 150 BCE – 75 CE\n.\n[\n5\n]\nThe modern development of\nerror correction codes\nis credited to\nRichard Hamming\nin 1947.\n[\n6\n]\nA description of\nHamming's code\nappeared in\nClaude Shannon\n's\nA Mathematical Theory of Communication\n[\n7\n]\nand was quickly generalized by\nMarcel J. E. Golay\n.\n[\n8\n]\nAll error-detection and correction schemes add some\nredundancy\n(i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted. Error detection and correction schemes can be either\nsystematic\nor non-systematic. In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of\ncheck bits\n(or\nparity data\n), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message.\nGood error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common\nchannel models\ninclude\nmemoryless\nmodels where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in\nbursts\n. Consequently, error-detecting and -correcting codes can be generally distinguished between\nrandom-error-detecting\/correcting\nand\nburst-error-detecting\/correcting\n. Some codes can also be suitable for a mixture of random errors and burst errors.\nIf the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as\nautomatic repeat request\n(ARQ), and is most notably used in the Internet. An alternate approach for error control is\nhybrid automatic repeat request\n(HARQ), which is a combination of ARQ and error-correction coding.\nTypes of error correction\n[\nedit\n]\nThere are three major types of error correction:\n[\n9\n]\nAutomatic repeat request\n[\nedit\n]\nAutomatic repeat request\n(ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and\/or negative acknowledgment messages, and\ntimeouts\nto achieve reliable data transmission. An\nacknowledgment\nis a message sent by the receiver to indicate that it has correctly received a\ndata frame\n.\nUsually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e., within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions.\nThree types of ARQ protocols are\nStop-and-wait ARQ\n,\nGo-Back-N ARQ\n, and\nSelective Repeat ARQ\n.\nARQ is appropriate if the communication channel has varying or unknown\ncapacity\n, such as is the case on the Internet. However, ARQ requires the availability of a\nback channel\n, results in possibly increased\nlatency\ndue to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of\nnetwork congestion\ncan put a strain on the server and overall network capacity.\n[\n10\n]\nFor example, ARQ is used on shortwave radio data links in the form of\nARQ-E\n, or combined with multiplexing as\nARQ-M\n.\nForward error correction\n[\nedit\n]\nForward error correction\n(FEC) is a process of adding\nredundant data\nsuch as an\nerror-correcting code\n(ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a\nbackchannel\nis not required in forward error correction. Error-correcting codes are used in\nlower-layer\ncommunication such as\ncellular network\n, high-speed\nfiber-optic communication\nand\nWi-Fi\n,\n[\n11\n]\n[\n12\n]\nas well as for reliable storage in media such as\nflash memory\n,\nhard disk\nand\nRAM\n.\n[\n13\n]\nError-correcting codes are usually distinguished between\nconvolutional codes\nand\nblock codes\n:\nConvolutional codes\nare processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the\nViterbi decoder\nallows\noptimal decoding\n.\nBlock codes\nare processed on a\nblock-by-block\nbasis. Early examples of block codes are\nrepetition codes\n,\nHamming codes\nand\nmultidimensional parity-check codes\n. They were followed by a number of efficient codes,\nReed–Solomon codes\nbeing the most notable due to their current widespread use.\nTurbo codes\nand\nlow-density parity-check codes\n(LDPC) are relatively new constructions that can provide almost\noptimal efficiency\n.\nShannon's theorem\nis an important theorem in forward error correction, and describes the maximum\ninformation rate\nat which reliable communication is possible over a channel that has a certain error probability or\nsignal-to-noise ratio\n(SNR). This strict upper limit is expressed in terms of the\nchannel capacity\n. More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a\ndiscrete memoryless channel\ncan be made arbitrarily small, provided that the\ncode rate\nis smaller than the channel capacity. The code rate is defined as the fraction\nk\/n\nof\nk\nsource symbols and\nn\nencoded symbols.\nThe actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have\nefficient\nencoding and decoding algorithms.\nHybrid ARQ\nis a combination of ARQ and forward error correction. There are two basic approaches:\n[\n10\n]\nMessages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check).\nMessages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message.\nThe latter approach is particularly attractive on an\nerasure channel\nwhen using a\nrateless erasure code\n.\nTypes of error detection\n[\nedit\n]\nError detection is most commonly realized using a suitable\nhash function\n(or specifically, a\nchecksum\n,\ncyclic redundancy check\nor other algorithm). A hash function adds a fixed-length\ntag\nto a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided.\nThere exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check's performance in detecting\nburst errors\n).\nMinimum distance coding\n[\nedit\n]\nA random-error-correcting code based on\nminimum distance coding\ncan provide a strict guarantee on the number of detectable errors, but it may not protect against a\npreimage attack\n.\nA\nrepetition code\nis a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern\n1011\n, the four-bit block can be repeated three times, thus producing\n1011 1011 1011\n. If this twelve-bit pattern was received as\n1010 1011 1011\n– where the first block is unlike the other two – an error has occurred.\nA repetition code is very inefficient and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g.,\n1010 1010 1010\nin the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of\nnumbers stations\n.\n[\n14\n]\n[\n15\n]\nA\nparity bit\nis a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous.\nParity bits added to each\nword\nsent are called\ntransverse redundancy checks\n, while those added at the end of a stream of\nwords\nare called\nlongitudinal redundancy checks\n.  For example, if each of a series of m-bit\nwords\nhas a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words.  With more error correction bits, more errors can be detected and in some cases corrected.\nThere are also other bit-grouping techniques.\nA\nchecksum\nof a message is a\nmodular arithmetic\nsum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a\nones'-complement\noperation prior to transmission to detect unintentional all-zero messages.\nChecksum schemes include parity bits,\ncheck digits\n, and\nlongitudinal redundancy checks\n. Some checksum schemes, such as the\nDamm algorithm\n, the\nLuhn algorithm\n, and the\nVerhoeff algorithm\n, are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers.\nCyclic redundancy check\n[\nedit\n]\nA\ncyclic redundancy check\n(CRC) is a non-secure\nhash function\ndesigned to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a\ngenerator polynomial\n, which is used as the\ndivisor\nin a\npolynomial long division\nover a\nfinite field\n, taking the input data as the\ndividend\n. The\nremainder\nbecomes the result.\nA CRC has properties that make it well suited for detecting\nburst errors\n. CRCs are particularly easy to implement in hardware and are therefore commonly used in\ncomputer networks\nand storage devices such as\nhard disk drives\n.\nThe parity bit can be seen as a special-case 1-bit CRC.\nCryptographic hash function\n[\nedit\n]\nThe output of a\ncryptographic hash function\n, also known as a\nmessage digest\n, can provide strong assurances about\ndata integrity\n, whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a\nkeyed hash\nor\nmessage authentication code\n(MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message.\nDigital signatures can provide strong assurances about data integrity, whether the changes of the data are accidental or maliciously introduced.\nDigital signatures are perhaps most notable for being part of the HTTPS protocol for securely browsing the web.\nError correction code\n[\nedit\n]\nAny error-correcting code can be used for error detection. A code with\nminimum\nHamming distance\n,\nd\n, can detect up to\nd\n− 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired.\nCodes with minimum Hamming distance\nd\n= 2 are degenerate cases of error-correcting codes and can be used to detect single errors. The parity bit is an example of a single-error-detecting code.\nApplications that require low latency (such as telephone conversations) cannot use\nautomatic repeat request\n(ARQ); they must use\nforward error correction\n(FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be usable.\nApplications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available.\nApplications that use ARQ must have a\nreturn channel\n; applications having no return channel cannot use ARQ. Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC.\nReliability and inspection engineering also make use of the theory of error-correcting codes,\n[\n16\n]\nas well as natural language.\n[\n17\n]\nIn a typical\nTCP\/IP\nstack, error control is performed at multiple levels:\nEach\nEthernet frame\nuses\nCRC-32\nerror detection. Frames with detected errors are discarded by the receiver hardware.\nThe\nIPv4\nheader contains a\nchecksum\nprotecting the contents of the header.\nPackets\nwith incorrect checksums are dropped within the network or at the receiver.\nThe checksum was omitted from the\nIPv6\nheader in order to minimize processing costs in\nnetwork routing\nand because current\nlink layer\ntechnology is assumed to provide sufficient error detection (see also RFC 3819).\nUDP\nhas an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the\nnetwork stack\n. The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection.\nTCP\nprovides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack and eventually get retransmitted using ARQ, either explicitly (such as through\nthree-way handshake\n) or implicitly due to a\ntimeout\n.\nDeep-space telecommunications\n[\nedit\n]\nThe development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded)\nconvolutional codes\nand\nReed–Muller codes\n.\n[\n18\n]\nThe Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a\nbell curve\n), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977.\nThe\nVoyager 1\nand\nVoyager 2\nmissions, which started in 1977, were designed to deliver color imaging and scientific information from\nJupiter\nand\nSaturn\n.\n[\n19\n]\nThis resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally\nViterbi-decoded\n) convolutional codes that could be\nconcatenated\nwith an outer\nGolay (24,12,8) code\n. The Voyager 2 craft additionally supported an implementation of a\nReed–Solomon code\n. The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft's extended journey to\nUranus\nand\nNeptune\n. After ECC system upgrades in 1989, both crafts used V2 RSV coding.\nThe\nConsultative Committee for Space Data Systems\ncurrently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as\nTurbo codes\nor\nLDPC codes\n.\nThe different kinds of deep space and orbital missions that are conducted suggest that trying to find a one-size-fits-all error correction system will be an ongoing problem. For missions close to Earth, the nature of the\nnoise\nin the\ncommunication channel\nis different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult.\nSatellite broadcasting\n[\nedit\n]\nThe demand for satellite\ntransponder\nbandwidth continues to grow, fueled by the desire to deliver television (including new channels and\nhigh-definition television\n) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected\nmodulation\nscheme and the proportion of capacity consumed by FEC.\nError detection and correction codes are often used to improve the reliability of data storage media.\n[\n20\n]\nA parity track capable of detecting single-bit errors was present on the first\nmagnetic tape data storage\nin 1951. The\noptimal rectangular code\nused in\ngroup coded recording\ntapes not only detects but also corrects single-bit errors. Some\nfile formats\n, particularly\narchive formats\n, include a checksum (most often CRC-32]) to detect corruption and truncation and can employ redundancy or\nparity files\nto recover portions of corrupted data.\nReed-Solomon codes\nare used in\ncompact discs\nto correct errors caused by scratches.\nModern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors.\n[\n21\n]\nRAID\nsystems use a variety of error correction techniques to recover data when a hard drive completely fails.  Filesystems such as\nZFS\nor\nBtrfs\n, as well as some\nRAID\nimplementations, support\ndata scrubbing\nand resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used.\n[\n22\n]\nThe recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware.\nError-correcting memory\n[\nedit\n]\nDynamic random-access memory\n(DRAM) may provide stronger protection against\nsoft errors\nby relying on error-correcting codes. Such error-correcting memory, known as\nECC\nor\nEDAC-protected\nmemory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased\nradiation\nin space.\nError-correcting memory controllers traditionally use\nHamming codes\n, although some use\ntriple modular redundancy\n.\nInterleaving\nallows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a\nsingle-event upset\n(SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained.\n[\n23\n]\nIn addition to hardware providing features required for ECC memory to operate,\noperating systems\nusually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the\nLinux kernel\n's\nEDAC\nsubsystem (previously known as\nBluesmoke\n), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the\nPCI bus\n.\n[\n24\n]\n[\n25\n]\n[\n26\n]\nA few systems\n[\nspecify\n]\nalso support\nmemory scrubbing\nto catch and correct errors early before they become unrecoverable.\nBerger code\nBurst error-correcting code\nECC memory\n, a type of computer data storage\nLink adaptation\nList of algorithms § Error detection and correction\nList of hash functions\n^\na\nb\n\"Masorah\".\nJewish Encyclopedia\n.\n^\nPratico, Gary D.; Pelt, Miles V. Van (2009).\nBasics of Biblical Hebrew Grammar: Second Edition\n. Zondervan.\nISBN\n \n978-0-310-55882-8\n.\n^\nMounce, William D. (2007).\nGreek for the Rest of Us: Using Greek Tools Without Mastering Biblical Languages\n. Zondervan. p. 289.\nISBN\n \n978-0-310-28289-1\n.\n^\nMishneh Torah\n, Tefillin, Mezuzah, and Sefer Torah, 1:2. Example English translation:\nEliyahu Touger\n.\nThe Rambam's Mishneh Torah\n.\nMoznaim Publishing Corporation\n.\n^\nBrian M. Fagan (5 December 1996). \"Dead Sea Scrolls\".\nThe Oxford Companion to Archaeology\n.\nOxford University Press\n.\nISBN\n \n0195076184\n.\n^\nThompson, Thomas M. (1983),\nFrom Error-Correcting Codes through Sphere Packings to Simple Groups\n, The Carus Mathematical Monographs (#21), The Mathematical Association of America, p. vii,\nISBN\n \n0-88385-023-0\n^\nShannon, C.E. (1948), \"A Mathematical Theory of Communication\",\nBell System Technical Journal\n,\n27\n(3):\n379–\n423,\ndoi\n:\n10.1002\/j.1538-7305.1948.tb01338.x\n,\nhdl\n:\n10338.dmlcz\/101429\n,\nPMID\n \n9230594\n^\nGolay, Marcel J. E. (1949), \"Notes on Digital Coding\",\nProc.I.R.E. (I.E.E.E.)\n,\n37\n: 657\n^\nGupta, Vikas; Verma, Chanderkant (November 2012). \"Error Detection and Correction: An Introduction\".\nInternational Journal of Advanced Research in Computer Science and Software Engineering\n.\n2\n(11).\nS2CID\n \n17499858\n.\n^\na\nb\nA. J. McAuley,\nReliable Broadband Communication Using a Burst Erasure Correcting Code\n, ACM SIGCOMM, 1990.\n^\nShah, Pradeep M.; Vyavahare, Prakash D.; Jain, Anjana (September 2015).\n\"Modern error correcting codes for 4G and beyond: Turbo codes and LDPC codes\"\n.\n2015 Radio and Antenna Days of the Indian Ocean (RADIO)\n. pp. \n1–\n2.\ndoi\n:\n10.1109\/RADIO.2015.7323369\n.\nISBN\n \n978-9-9903-7339-4\n.\nS2CID\n \n28885076\n. Retrieved\n22 May\n2022\n.\n^\n\"IEEE SA - IEEE 802.11ac-2013\"\n.\nIEEE Standards Association\n. Archived from\nthe original\non 2022-05-22\n. Retrieved\n2022-05-22\n.\n^\n\"Transition to Advanced Format 4K Sector Hard Drives | Seagate US\"\n.\nSeagate.com\n. Retrieved\n22 May\n2022\n.\n^\nFrank van Gerwen.\n\"Numbers (and other mysterious) stations\"\n. Archived from\nthe original\non 12 July 2017\n. Retrieved\n12 March\n2012\n.\n^\nGary Cutlack (25 August 2010).\n\"Mysterious Russian 'Numbers Station' Changes Broadcast After 20 Years\"\n.\nGizmodo\n. Archived from\nthe original\non 5 July 2017\n. Retrieved\n12 March\n2012\n.\n^\nBen-Gal I.; Herer Y.; Raz T. (2003).\n\"Self-correcting inspection procedure under inspection errors\"\n(PDF)\n.\nIIE Transactions\n. IIE Transactions on Quality and Reliability, 34(6), pp. 529-540. Archived from\nthe original\n(PDF)\non 2013-10-13\n. Retrieved\n2014-01-10\n.\n^\nYvo Meeres, Tommi A. Pirinen (2021). \"Vowel Harmony Viewed as Error-Correcting Code\".\nProceedings of the Society for Computation in Linguistics\n.\n4\n(1):\n313–\n322.\ndoi\n:\n10.7275\/bjvb-2n37\n.\n^\nK. Andrews et al.,\nThe Development of Turbo and LDPC Codes for Deep-Space Applications\n, Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007.\n^\nHuffman, William Cary;\nPless, Vera S.\n(2003).\nFundamentals of Error-Correcting Codes\n.\nCambridge University Press\n.\nISBN\n \n978-0-521-78280-7\n.\n^\nKurtas, Erozan M.; Vasic, Bane (2018-10-03).\nAdvanced Error Control Techniques for Data Storage Systems\n. CRC Press.\nISBN\n \n978-1-4200-3649-7\n.\n^\nScott A. Moulton.\n\"My Hard Drive Died\"\n. Archived from\nthe original\non 2008-02-02.\n^\nQiao, Zhi; Fu, Song; Chen, Hsing-Bung; Settlemyer, Bradley (2019). \"Building Reliable High-Performance Storage Systems: An Empirical and Analytical Study\".\n2019 IEEE International Conference on Cluster Computing (CLUSTER)\n. pp. \n1–\n10.\ndoi\n:\n10.1109\/CLUSTER.2019.8891006\n.\nISBN\n \n978-1-7281-4734-5\n.\nS2CID\n \n207951690\n.\n^\n\"Using StrongArm SA-1110 in the On-Board Computer of Nanosatellite\"\n. Tsinghua Space Center,\nTsinghua University\n, Beijing. Archived from\nthe original\non 2011-10-02\n. Retrieved\n2009-02-16\n.\n^\nJeff Layton.\n\"Error Detection and Correction\"\n.\nLinux Magazine\n. Retrieved\n2014-08-12\n.\n^\n\"EDAC Project\"\n.\nbluesmoke.sourceforge.net\n. Retrieved\n2014-08-12\n.\n^\n\"Documentation\/edac.txt\"\n.\nLinux kernel documentation\n.\nkernel.org\n. 2014-06-16. Archived from\nthe original\non 2009-09-05\n. Retrieved\n2014-08-12\n.\nShu Lin; Daniel J. Costello, Jr. (1983).\nError Control Coding: Fundamentals and Applications\n.\nPrentice Hall\n.\nISBN\n \n0-13-283796-X\n.\nSoftECC: A System for Software Memory Integrity Checking\nA Tunable, Software-based DRAM Error Detection and Correction Library for HPC\nArchived\n2014-11-07 at the\nWayback Machine\nDetection and Correction of Silent Data Corruption for Large-Scale High-Performance Computing\nArchived\n2014-11-07 at the\nWayback Machine\nThe on-line textbook: Information Theory, Inference, and Learning Algorithms\n, by\nDavid J.C. MacKay\n, contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including\nlow-density parity-check codes\n,\nturbo codes\n, and\nfountain codes\n.\nECC Page\n– implementations of popular ECC encoding and decoding routines","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages \"A list of all special pages [q]\")\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/enwiki-25.svg) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en-25.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Error+detection+and+correction \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Error+detection+and+correction \"You're encouraged to log in; however, it's not mandatory. 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[o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction)\n- [1 Definitions](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Definitions)\n- [2 History](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#History)\n- [3 Principles](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Principles)\n- [4 Types of error correction](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Types_of_error_correction)\n  Toggle Types of error correction subsection\n  - [4\\.1 Automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Automatic_repeat_request)\n  - [4\\.2 Forward error correction](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Forward_error_correction)\n  - [4\\.3 Hybrid schemes](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Hybrid_schemes)\n- [5 Types of error detection](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Types_of_error_detection)\n  Toggle Types of error detection subsection\n  - [5\\.1 Minimum distance coding](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Minimum_distance_coding)\n  - [5\\.2 Repetition codes](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Repetition_codes)\n  - [5\\.3 Parity bit](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Parity_bit)\n  - [5\\.4 Checksum](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Checksum)\n  - [5\\.5 Cyclic redundancy check](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Cyclic_redundancy_check)\n  - [5\\.6 Cryptographic hash function](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Cryptographic_hash_function)\n  - [5\\.7 Digital signature](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Digital_signature)\n  - [5\\.8 Error correction code](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Error_correction_code)\n- [6 Applications](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Applications)\n  Toggle Applications subsection\n  - [6\\.1 Internet](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Internet)\n  - [6\\.2 Deep-space telecommunications](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Deep-space_telecommunications)\n  - [6\\.3 Satellite broadcasting](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Satellite_broadcasting)\n  - [6\\.4 Data storage](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Data_storage)\n  - [6\\.5 Error-correcting memory](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Error-correcting_memory)\n- [7 See also](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#See_also)\n- [8 References](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#References)\n- [9 Further reading](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#Further_reading)\n- [10 External links](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#External_links)\n\nToggle the table of contents\n\n# Error detection and correction\n36 languages\n\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D9%83%D8%B4%D9%81_%D8%A7%D9%84%D8%A3%D8%AE%D8%B7%D8%A7%D8%A1_%D9%88%D8%AA%D8%B5%D8%AD%D9%8A%D8%AD%D9%87%D8%A7 \"كشف الأخطاء وتصحيحها – Arabic\")\n- [Asturianu](https:\/\/ast.wikipedia.org\/wiki\/Deteici%C3%B3n_y_correici%C3%B3n_d%27errores \"Deteición y correición d'errores – Asturian\")\n- [Azərbaycanca](https:\/\/az.wikipedia.org\/wiki\/X%C9%99ta_a%C5%9Fkarlama_kodla%C5%9Fd%C4%B1rmas%C4%B1 \"Xəta aşkarlama kodlaşdırması – Azerbaijani\")\n- [Беларуская (тарашкевіца)](https:\/\/be-tarask.wikipedia.org\/wiki\/%D0%9A%D0%B0%D1%80%D1%8D%D0%BA%D1%86%D1%8B%D1%8F_%D0%BF%D0%B0%D0%BC%D1%8B%D0%BB%D0%B0%D0%BA \"Карэкцыя памылак – Belarusian (Taraškievica orthography)\")\n- [Беларуская](https:\/\/be.wikipedia.org\/wiki\/%D0%92%D1%8B%D1%8F%D1%9E%D0%BB%D0%B5%D0%BD%D0%BD%D0%B5_%D1%96_%D0%B2%D1%8B%D0%BF%D1%80%D0%B0%D1%9E%D0%BB%D0%B5%D0%BD%D0%BD%D0%B5_%D0%BF%D0%B0%D0%BC%D1%8B%D0%BB%D0%B0%D0%BA \"Выяўленне і выпраўленне памылак – Belarusian\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Detector_i_corrector_d%27errors \"Detector i corrector d'errors – Catalan\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/Detekce_a_oprava_chyb \"Detekce a oprava chyb – Czech\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Fehlerkorrekturverfahren \"Fehlerkorrekturverfahren – German\")\n- [Ελληνικά](https:\/\/el.wikipedia.org\/wiki\/%CE%91%CE%BD%CE%AF%CF%87%CE%BD%CE%B5%CF%85%CF%83%CE%B7_%CE%BA%CE%B1%CE%B9_%CE%B4%CE%B9%CF%8C%CF%81%CE%B8%CF%89%CF%83%CE%B7_%CF%83%CF%86%CE%B1%CE%BB%CE%BC%CE%AC%CF%84%CF%89%CE%BD \"Ανίχνευση και διόρθωση σφαλμάτων – Greek\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Detecci%C3%B3n_y_correcci%C3%B3n_de_errores \"Detección y corrección de errores – Spanish\")\n- [Euskara](https:\/\/eu.wikipedia.org\/wiki\/Errore-detekzio \"Errore-detekzio – Basque\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D8%AA%D8%B4%D8%AE%DB%8C%D8%B5_%D9%88_%D8%AA%D8%B5%D8%AD%DB%8C%D8%AD_%D8%AE%D8%B7%D8%A7 \"تشخیص و تصحیح خطا – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Virheen_havaitseminen_ja_korjaaminen \"Virheen havaitseminen ja korjaaminen – Finnish\")\n- [हिन्दी](https:\/\/hi.wikipedia.org\/wiki\/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A5%81%E0%A4%9F%E0%A4%BF_%E0%A4%B8%E0%A4%82%E0%A4%B8%E0%A5%82%E0%A4%9A%E0%A4%A8_%E0%A4%94%E0%A4%B0_%E0%A4%B8%E0%A5%81%E0%A4%A7%E0%A4%BE%E0%A4%B0 \"त्रुटि संसूचन और सुधार – Hindi\")\n- [Magyar](https:\/\/hu.wikipedia.org\/wiki\/Hibajav%C3%ADt%C3%A1s \"Hibajavítás – Hungarian\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Rilevazione_e_correzione_d%27errore \"Rilevazione e correzione d'errore – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E8%AA%A4%E3%82%8A%E6%A4%9C%E5%87%BA%E8%A8%82%E6%AD%A3 \"誤り検出訂正 – Japanese\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EC%98%A4%EB%A5%98_%EA%B2%80%EC%B6%9C_%EC%A0%95%EC%A0%95 \"오류 검출 정정 – Korean\")\n- [Latina](https:\/\/la.wikipedia.org\/wiki\/Deprehensio_et_correctio_mendorum \"Deprehensio et correctio mendorum – Latin\")\n- [Latviešu](https:\/\/lv.wikipedia.org\/wiki\/K%C4%BC%C5%ABdu_kontrole \"Kļūdu kontrole – Latvian\")\n- [Nederlands](https:\/\/nl.wikipedia.org\/wiki\/Foutdetectie_en_-correctie \"Foutdetectie en -correctie – Dutch\")\n- [Norsk nynorsk](https:\/\/nn.wikipedia.org\/wiki\/Feildeteksjon_og_korreksjon \"Feildeteksjon og korreksjon – Norwegian Nynorsk\")\n- [Polski](https:\/\/pl.wikipedia.org\/wiki\/Kodowanie_korekcyjne \"Kodowanie korekcyjne – Polish\")\n- [Piemontèis](https:\/\/pms.wikipedia.org\/wiki\/Rilevament_e_coression_eror \"Rilevament e coression eror – Piedmontese\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/Detec%C3%A7%C3%A3o_e_corre%C3%A7%C3%A3o_de_erros \"Detecção e correção de erros – Portuguese\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%9A%D0%BE%D0%BD%D1%82%D1%80%D0%BE%D0%BB%D1%8C_%D0%BE%D1%88%D0%B8%D0%B1%D0%BE%D0%BA \"Контроль ошибок – Russian\")\n- [Simple English](https:\/\/simple.wikipedia.org\/wiki\/Error_detection_and_correction \"Error detection and correction – Simple English\")\n- [Српски \/ srpski](https:\/\/sr.wikipedia.org\/wiki\/Detekcija_i_korekcija_gre%C5%A1aka \"Detekcija i korekcija grešaka – Serbian\")\n- [Svenska](https:\/\/sv.wikipedia.org\/wiki\/Felkorrigering \"Felkorrigering – Swedish\")\n- [Kiswahili](https:\/\/sw.wikipedia.org\/wiki\/Ugunduzi_na_usahihishaji_wa_makosa \"Ugunduzi na usahihishaji wa makosa – Swahili\")\n- [ไทย](https:\/\/th.wikipedia.org\/wiki\/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%95%E0%B8%A3%E0%B8%A7%E0%B8%88%E0%B8%AB%E0%B8%B2%E0%B9%81%E0%B8%A5%E0%B8%B0%E0%B9%81%E0%B8%81%E0%B9%89%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%9C%E0%B8%B4%E0%B8%94%E0%B8%9E%E0%B8%A5%E0%B8%B2%E0%B8%94 \"การตรวจหาและแก้ความผิดพลาด – Thai\")\n- [Türkçe](https:\/\/tr.wikipedia.org\/wiki\/Hata_bulma_ve_d%C3%BCzeltme \"Hata bulma ve düzeltme – Turkish\")\n- [Українська](https:\/\/uk.wikipedia.org\/wiki\/%D0%92%D0%B8%D1%8F%D0%B2%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F_%D1%82%D0%B0_%D0%B2%D0%B8%D0%BF%D1%80%D0%B0%D0%B2%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F_%D0%BF%D0%BE%D0%BC%D0%B8%D0%BB%D0%BE%D0%BA \"Виявлення та виправлення помилок – Ukrainian\")\n- [اردو](https:\/\/ur.wikipedia.org\/wiki\/%D8%BA%D9%84%D8%B7%DB%8C_%DA%A9%DB%8C_%D8%B4%D9%86%D8%A7%D8%AE%D8%AA_%D8%A7%D9%88%D8%B1_%D8%AF%D8%B1%D8%B3%D8%AA%DB%8C \"غلطی کی شناخت اور درستی – Urdu\")\n- [粵語](https:\/\/zh-yue.wikipedia.org\/wiki\/%E9%8C%AF%E8%AA%A4%E6%AA%A2%E6%B8%AC%E5%90%8C%E7%B3%BE%E6%AD%A3 \"錯誤檢測同糾正 – Cantonese\")\n- [中文](https:\/\/zh.wikipedia.org\/wiki\/%E9%94%99%E8%AF%AF%E6%A3%80%E6%B5%8B%E4%B8%8E%E7%BA%A0%E6%AD%A3 \"错误检测与纠正 – Chinese\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q1062839#sitelinks-wikipedia \"Edit interlanguage links\")\n\n- [Article](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction \"View the content page [c]\")\n- [Talk](https:\/\/en.wikipedia.org\/wiki\/Talk:Error_detection_and_correction \"Discuss improvements to the content page [t]\")\n\nEnglish\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=history \"Past revisions of this page [h]\")\n\nTools\n\nTools\n\nmove to sidebar\n\nhide\n\nActions\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=history)\n\nGeneral\n\n- [What links here](https:\/\/en.wikipedia.org\/wiki\/Special:WhatLinksHere\/Error_detection_and_correction \"List of all 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URL](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FError_detection_and_correction)\n\nPrint\/export\n\n- [Download as PDF](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:DownloadAsPdf&page=Error_detection_and_correction&action=show-download-screen \"Download this page as a PDF file\")\n- [Printable version](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q1062839 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nReliable digital data delivery methods on unreliable channels\n\nNot to be confused with [Exception handling](https:\/\/en.wikipedia.org\/wiki\/Exception_handling \"Exception handling\").\n\n|   |   |\n|---|---|\n| [![icon](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/9\/99\/Question_book-new.svg\/60px-Question_book-new.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Question_book-new.svg) | This article **needs additional citations for [verification](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Verifiability \"Wikipedia:Verifiability\")**. Please help [improve this article](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Error_detection_and_correction \"Special:EditPage\/Error detection and correction\") by [adding citations to reliable sources](https:\/\/en.wikipedia.org\/wiki\/Help:Referencing_for_beginners \"Help:Referencing for beginners\") in this article. Unsourced material may be challenged and removed. *Find sources:* [\"Error detection and correction\"](https:\/\/www.google.com\/search?as_eq=wikipedia&q=%22Error+detection+and+correction%22) – [news](https:\/\/www.google.com\/search?tbm=nws&q=%22Error+detection+and+correction%22+-wikipedia&tbs=ar:1) **·** [newspapers](https:\/\/www.google.com\/search?&q=%22Error+detection+and+correction%22&tbs=bkt:s&tbm=bks) **·** [books](https:\/\/www.google.com\/search?tbs=bks:1&q=%22Error+detection+and+correction%22+-wikipedia) **·** [scholar](https:\/\/scholar.google.com\/scholar?q=%22Error+detection+and+correction%22) **·** [JSTOR](https:\/\/www.jstor.org\/action\/doBasicSearch?Query=%22Error+detection+and+correction%22&acc=on&wc=on) *(August 2008)* *([Learn how and when to remove this message](https:\/\/en.wikipedia.org\/wiki\/Help:Maintenance_template_removal \"Help:Maintenance template removal\"))* |\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/ef\/Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg\/250px-Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg)\n\nTo clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied [Reed–Solomon error correction](https:\/\/en.wikipedia.org\/wiki\/Reed%E2%80%93Solomon_error_correction \"Reed–Solomon error correction\") (right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted.\n\nIn [information theory](https:\/\/en.wikipedia.org\/wiki\/Information_theory \"Information theory\") and [coding theory](https:\/\/en.wikipedia.org\/wiki\/Coding_theory \"Coding theory\") with applications in [computer science](https:\/\/en.wikipedia.org\/wiki\/Computer_science \"Computer science\") and [telecommunications](https:\/\/en.wikipedia.org\/wiki\/Telecommunications \"Telecommunications\"), **error detection and correction** (**EDAC**) or **error control** are techniques that enable [reliable delivery](https:\/\/en.wikipedia.org\/wiki\/Reliable_delivery \"Reliable delivery\") of [digital data](https:\/\/en.wikipedia.org\/wiki\/Digital_data \"Digital data\") over unreliable [communication channels](https:\/\/en.wikipedia.org\/wiki\/Communication_channel \"Communication channel\"). Many communication channels are subject to [channel noise](https:\/\/en.wikipedia.org\/wiki\/Channel_noise \"Channel noise\"), and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases.\n\n## Definitions\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=1 \"Edit section: Definitions\")\\]\n\n*Error detection* is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver.\n\n*Error correction* is the detection of errors and reconstruction of the original, error-free data.\n\n## History\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=2 \"Edit section: History\")\\]\n\nIn classical antiquity, [copyists](https:\/\/en.wikipedia.org\/wiki\/Copyist \"Copyist\") of the [Hebrew Bible](https:\/\/en.wikipedia.org\/wiki\/Hebrew_Bible \"Hebrew Bible\") were paid for their work according to the number of [stichs](https:\/\/en.wiktionary.org\/wiki\/stich \"wiktionary:stich\") (lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-Jewish-1) This also helped ensure accuracy in the transmission of the text with the production of subsequent copies.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-2)[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-3) Between the 7th and 10th centuries CE a [group of Jewish scribes](https:\/\/en.wikipedia.org\/wiki\/Masoretes \"Masoretes\") formalized and expanded this to create the [Numerical Masorah](https:\/\/en.wikipedia.org\/wiki\/Masoretic_Text#Numerical_Masorah \"Masoretic Text\") to ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-Jewish-1) Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-4) The effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the [Dead Sea Scrolls](https:\/\/en.wikipedia.org\/wiki\/Dead_Sea_Scrolls \"Dead Sea Scrolls\") in 1947–1956, dating from c. 150 BCE – 75 CE.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-5)\n\nThe modern development of [error correction codes](https:\/\/en.wikipedia.org\/wiki\/Error_correction_code \"Error correction code\") is credited to [Richard Hamming](https:\/\/en.wikipedia.org\/wiki\/Richard_Hamming \"Richard Hamming\") in 1947.[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-Thompson-6) A description of [Hamming's code](https:\/\/en.wikipedia.org\/wiki\/Hamming%27s_code \"Hamming's code\") appeared in [Claude Shannon](https:\/\/en.wikipedia.org\/wiki\/Claude_Shannon \"Claude Shannon\")'s *A Mathematical Theory of Communication*[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-7) and was quickly generalized by [Marcel J. E. Golay](https:\/\/en.wikipedia.org\/wiki\/Marcel_J._E._Golay \"Marcel J. E. Golay\").[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-8)\n\n## Principles\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=3 \"Edit section: Principles\")\\]\n\nAll error-detection and correction schemes add some [redundancy](https:\/\/en.wikipedia.org\/wiki\/Redundancy_\\(information_theory\\) \"Redundancy (information theory)\") (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted. Error detection and correction schemes can be either [systematic](https:\/\/en.wikipedia.org\/wiki\/Systematic_code \"Systematic code\") or non-systematic. In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of *check bits* (or *parity data*), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message.\n\nGood error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common [channel models](https:\/\/en.wikipedia.org\/wiki\/Channel_model \"Channel model\") include [memoryless](https:\/\/en.wikipedia.org\/wiki\/Memoryless \"Memoryless\") models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in [bursts](https:\/\/en.wikipedia.org\/wiki\/Burst_error \"Burst error\"). Consequently, error-detecting and -correcting codes can be generally distinguished between *random-error-detecting\/correcting* and *burst-error-detecting\/correcting*. Some codes can also be suitable for a mixture of random errors and burst errors.\n\nIf the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as [automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Automatic_repeat_request \"Automatic repeat request\") (ARQ), and is most notably used in the Internet. An alternate approach for error control is [hybrid automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Hybrid_automatic_repeat_request \"Hybrid automatic repeat request\") (HARQ), which is a combination of ARQ and error-correction coding.\n\n## Types of error correction\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=4 \"Edit section: Types of error correction\")\\]\n\nThere are three major types of error correction:[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-9)\n\n### Automatic repeat request\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=5 \"Edit section: Automatic repeat request\")\\]\n\n[Automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Automatic_repeat_request \"Automatic repeat request\") (ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and\/or negative acknowledgment messages, and [timeouts](https:\/\/en.wikipedia.org\/wiki\/Timeout_\\(computing\\) \"Timeout (computing)\") to achieve reliable data transmission. An *acknowledgment* is a message sent by the receiver to indicate that it has correctly received a [data frame](https:\/\/en.wikipedia.org\/wiki\/Data_frame \"Data frame\").\n\nUsually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e., within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions.\n\nThree types of ARQ protocols are [Stop-and-wait ARQ](https:\/\/en.wikipedia.org\/wiki\/Stop-and-wait_ARQ \"Stop-and-wait ARQ\"), [Go-Back-N ARQ](https:\/\/en.wikipedia.org\/wiki\/Go-Back-N_ARQ \"Go-Back-N ARQ\"), and [Selective Repeat ARQ](https:\/\/en.wikipedia.org\/wiki\/Selective_Repeat_ARQ \"Selective Repeat ARQ\").\n\nARQ is appropriate if the communication channel has varying or unknown [capacity](https:\/\/en.wikipedia.org\/wiki\/Channel_capacity \"Channel capacity\"), such as is the case on the Internet. However, ARQ requires the availability of a [back channel](https:\/\/en.wikipedia.org\/wiki\/Backward_channel \"Backward channel\"), results in possibly increased [latency](https:\/\/en.wikipedia.org\/wiki\/Network_latency \"Network latency\") due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of [network congestion](https:\/\/en.wikipedia.org\/wiki\/Network_congestion \"Network congestion\") can put a strain on the server and overall network capacity.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-reliable-erasure-code-10)\n\nFor example, ARQ is used on shortwave radio data links in the form of [ARQ-E](https:\/\/en.wikipedia.org\/wiki\/ARQ-E \"ARQ-E\"), or combined with multiplexing as [ARQ-M](https:\/\/en.wikipedia.org\/wiki\/ARQ-M \"ARQ-M\").\n\n### Forward error correction\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=6 \"Edit section: Forward error correction\")\\]\n\n[Forward error correction](https:\/\/en.wikipedia.org\/wiki\/Forward_error_correction \"Forward error correction\") (FEC) is a process of adding [redundant data](https:\/\/en.wikipedia.org\/wiki\/Redundancy_\\(information_theory\\) \"Redundancy (information theory)\") such as an [error-correcting code](https:\/\/en.wikipedia.org\/wiki\/Error-correcting_code \"Error-correcting code\") (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a [backchannel](https:\/\/en.wikipedia.org\/wiki\/Backchannel \"Backchannel\") is not required in forward error correction. Error-correcting codes are used in [lower-layer](https:\/\/en.wikipedia.org\/wiki\/Physical_layer \"Physical layer\") communication such as [cellular network](https:\/\/en.wikipedia.org\/wiki\/Cellular_network \"Cellular network\"), high-speed [fiber-optic communication](https:\/\/en.wikipedia.org\/wiki\/Fiber-optic_communication \"Fiber-optic communication\") and [Wi-Fi](https:\/\/en.wikipedia.org\/wiki\/Wi-Fi \"Wi-Fi\"),[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-11)[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-12) as well as for reliable storage in media such as [flash memory](https:\/\/en.wikipedia.org\/wiki\/Flash_memory \"Flash memory\"), [hard disk](https:\/\/en.wikipedia.org\/wiki\/Hard_disk \"Hard disk\") and [RAM](https:\/\/en.wikipedia.org\/wiki\/ECC_memory \"ECC memory\").[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-13)\n\nError-correcting codes are usually distinguished between [convolutional codes](https:\/\/en.wikipedia.org\/wiki\/Convolutional_code \"Convolutional code\") and [block codes](https:\/\/en.wikipedia.org\/wiki\/Block_code \"Block code\"):\n\n- *Convolutional codes* are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the [Viterbi decoder](https:\/\/en.wikipedia.org\/wiki\/Viterbi_decoder \"Viterbi decoder\") allows [optimal decoding](https:\/\/en.wikipedia.org\/wiki\/Maximum_likelihood_decoding \"Maximum likelihood decoding\").\n- *Block codes* are processed on a [block-by-block](https:\/\/en.wikipedia.org\/wiki\/Block_\\(telecommunications\\) \"Block (telecommunications)\") basis. Early examples of block codes are [repetition codes](https:\/\/en.wikipedia.org\/wiki\/Repetition_code \"Repetition code\"), [Hamming codes](https:\/\/en.wikipedia.org\/wiki\/Hamming_code \"Hamming code\") and [multidimensional parity-check codes](https:\/\/en.wikipedia.org\/wiki\/Multidimensional_parity-check_code \"Multidimensional parity-check code\"). They were followed by a number of efficient codes, [Reed–Solomon codes](https:\/\/en.wikipedia.org\/wiki\/Reed%E2%80%93Solomon_code \"Reed–Solomon code\") being the most notable due to their current widespread use. [Turbo codes](https:\/\/en.wikipedia.org\/wiki\/Turbo_code \"Turbo code\") and [low-density parity-check codes](https:\/\/en.wikipedia.org\/wiki\/Low-density_parity-check_code \"Low-density parity-check code\") (LDPC) are relatively new constructions that can provide almost [optimal efficiency](https:\/\/en.wikipedia.org\/wiki\/Category:Capacity-approaching_codes \"Category:Capacity-approaching codes\").\n\n[Shannon's theorem](https:\/\/en.wikipedia.org\/wiki\/Shannon%27s_theorem \"Shannon's theorem\") is an important theorem in forward error correction, and describes the maximum [information rate](https:\/\/en.wikipedia.org\/wiki\/Information_rate \"Information rate\") at which reliable communication is possible over a channel that has a certain error probability or [signal-to-noise ratio](https:\/\/en.wikipedia.org\/wiki\/Signal-to-noise_ratio \"Signal-to-noise ratio\") (SNR). This strict upper limit is expressed in terms of the [channel capacity](https:\/\/en.wikipedia.org\/wiki\/Channel_capacity \"Channel capacity\"). More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a [discrete memoryless channel](https:\/\/en.wikipedia.org\/wiki\/Channel_model \"Channel model\") can be made arbitrarily small, provided that the [code rate](https:\/\/en.wikipedia.org\/wiki\/Code_rate \"Code rate\") is smaller than the channel capacity. The code rate is defined as the fraction *k\/n* of *k* source symbols and *n* encoded symbols.\n\nThe actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have [efficient](https:\/\/en.wikipedia.org\/wiki\/Polynomial_time \"Polynomial time\") encoding and decoding algorithms.\n\n### Hybrid schemes\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=7 \"Edit section: Hybrid schemes\")\\]\n\nMain article: [Hybrid ARQ](https:\/\/en.wikipedia.org\/wiki\/Hybrid_ARQ \"Hybrid ARQ\")\n\n[Hybrid ARQ](https:\/\/en.wikipedia.org\/wiki\/Hybrid_ARQ \"Hybrid ARQ\") is a combination of ARQ and forward error correction. There are two basic approaches:[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-reliable-erasure-code-10)\n\n- Messages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check).\n- Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message.\n\nThe latter approach is particularly attractive on an [erasure channel](https:\/\/en.wikipedia.org\/wiki\/Erasure_channel \"Erasure channel\") when using a [rateless erasure code](https:\/\/en.wikipedia.org\/wiki\/Rateless_erasure_code \"Rateless erasure code\").\n\n## Types of error detection\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=8 \"Edit section: Types of error detection\")\\]\n\nError detection is most commonly realized using a suitable [hash function](https:\/\/en.wikipedia.org\/wiki\/Hash_function \"Hash function\") (or specifically, a [checksum](https:\/\/en.wikipedia.org\/wiki\/Checksum \"Checksum\"), [cyclic redundancy check](https:\/\/en.wikipedia.org\/wiki\/Cyclic_redundancy_check \"Cyclic redundancy check\") or other algorithm). A hash function adds a fixed-length *tag* to a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided.\n\nThere exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check's performance in detecting [burst errors](https:\/\/en.wikipedia.org\/wiki\/Burst_error \"Burst error\")).\n\n### Minimum distance coding\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=9 \"Edit section: Minimum distance coding\")\\]\n\nA random-error-correcting code based on [minimum distance coding](https:\/\/en.wikipedia.org\/wiki\/Minimum_distance_coding \"Minimum distance coding\") can provide a strict guarantee on the number of detectable errors, but it may not protect against a [preimage attack](https:\/\/en.wikipedia.org\/wiki\/Preimage_attack \"Preimage attack\").\n\n### Repetition codes\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=10 \"Edit section: Repetition codes\")\\]\n\nA [repetition code](https:\/\/en.wikipedia.org\/wiki\/Repetition_code \"Repetition code\") is a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern `1011`, the four-bit block can be repeated three times, thus producing `1011 1011 1011`. If this twelve-bit pattern was received as `1010 1011 1011` – where the first block is unlike the other two – an error has occurred.\n\nA repetition code is very inefficient and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g., `1010 1010 1010` in the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of [numbers stations](https:\/\/en.wikipedia.org\/wiki\/Numbers_station \"Numbers station\").[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-14)[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-15)\n\n### Parity bit\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=11 \"Edit section: Parity bit\")\\]\n\nMain article: [Parity bit](https:\/\/en.wikipedia.org\/wiki\/Parity_bit \"Parity bit\")\n\nA *parity bit* is a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous.\n\nParity bits added to each *word* sent are called [transverse redundancy checks](https:\/\/en.wikipedia.org\/wiki\/Transverse_redundancy_check \"Transverse redundancy check\"), while those added at the end of a stream of *words* are called [longitudinal redundancy checks](https:\/\/en.wikipedia.org\/wiki\/Longitudinal_redundancy_check \"Longitudinal redundancy check\"). For example, if each of a series of m-bit *words* has a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words. With more error correction bits, more errors can be detected and in some cases corrected.\n\nThere are also other bit-grouping techniques.\n\n### Checksum\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=12 \"Edit section: Checksum\")\\]\n\nMain article: [Checksum](https:\/\/en.wikipedia.org\/wiki\/Checksum \"Checksum\")\n\nA *checksum* of a message is a [modular arithmetic](https:\/\/en.wikipedia.org\/wiki\/Modular_arithmetic \"Modular arithmetic\") sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a [ones'-complement](https:\/\/en.wikipedia.org\/wiki\/Ones%27-complement \"Ones'-complement\") operation prior to transmission to detect unintentional all-zero messages.\n\nChecksum schemes include parity bits, [check digits](https:\/\/en.wikipedia.org\/wiki\/Check_digit \"Check digit\"), and [longitudinal redundancy checks](https:\/\/en.wikipedia.org\/wiki\/Longitudinal_redundancy_check \"Longitudinal redundancy check\"). Some checksum schemes, such as the [Damm algorithm](https:\/\/en.wikipedia.org\/wiki\/Damm_algorithm \"Damm algorithm\"), the [Luhn algorithm](https:\/\/en.wikipedia.org\/wiki\/Luhn_algorithm \"Luhn algorithm\"), and the [Verhoeff algorithm](https:\/\/en.wikipedia.org\/wiki\/Verhoeff_algorithm \"Verhoeff algorithm\"), are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers.\n\n### Cyclic redundancy check\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=13 \"Edit section: Cyclic redundancy check\")\\]\n\nMain article: [Cyclic redundancy check](https:\/\/en.wikipedia.org\/wiki\/Cyclic_redundancy_check \"Cyclic redundancy check\")\n\nA *cyclic redundancy check* (CRC) is a non-secure [hash function](https:\/\/en.wikipedia.org\/wiki\/Hash_function \"Hash function\") designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a *generator polynomial*, which is used as the [divisor](https:\/\/en.wikipedia.org\/wiki\/Divisor \"Divisor\") in a [polynomial long division](https:\/\/en.wikipedia.org\/wiki\/Polynomial_long_division \"Polynomial long division\") over a [finite field](https:\/\/en.wikipedia.org\/wiki\/Finite_field \"Finite field\"), taking the input data as the [dividend](https:\/\/en.wikipedia.org\/wiki\/Dividend \"Dividend\"). The [remainder](https:\/\/en.wikipedia.org\/wiki\/Remainder \"Remainder\") becomes the result.\n\nA CRC has properties that make it well suited for detecting [burst errors](https:\/\/en.wikipedia.org\/wiki\/Burst_error \"Burst error\"). CRCs are particularly easy to implement in hardware and are therefore commonly used in [computer networks](https:\/\/en.wikipedia.org\/wiki\/Computer_network \"Computer network\") and storage devices such as [hard disk drives](https:\/\/en.wikipedia.org\/wiki\/Hard_disk_drives \"Hard disk drives\").\n\nThe parity bit can be seen as a special-case 1-bit CRC.\n\n### Cryptographic hash function\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=14 \"Edit section: Cryptographic hash function\")\\]\n\nMain article: [Cryptographic hash function](https:\/\/en.wikipedia.org\/wiki\/Cryptographic_hash_function \"Cryptographic hash function\")\n\nThe output of a *cryptographic hash function*, also known as a *message digest*, can provide strong assurances about [data integrity](https:\/\/en.wikipedia.org\/wiki\/Data_integrity \"Data integrity\"), whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a *keyed hash* or [message authentication code](https:\/\/en.wikipedia.org\/wiki\/Message_authentication_code \"Message authentication code\") (MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message.\n\n### Digital signature\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=15 \"Edit section: Digital signature\")\\]\n\nMain article: [Digital signature](https:\/\/en.wikipedia.org\/wiki\/Digital_signature \"Digital signature\")\n\nDigital signatures can provide strong assurances about data integrity, whether the changes of the data are accidental or maliciously introduced. Digital signatures are perhaps most notable for being part of the HTTPS protocol for securely browsing the web.\n\n### Error correction code\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=16 \"Edit section: Error correction code\")\\]\n\nMain article: [Error correction code](https:\/\/en.wikipedia.org\/wiki\/Error_correction_code \"Error correction code\")\n\nAny error-correcting code can be used for error detection. A code with *minimum [Hamming distance](https:\/\/en.wikipedia.org\/wiki\/Hamming_distance \"Hamming distance\")*, *d*, can detect up to *d* − 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired.\n\nCodes with minimum Hamming distance *d* = 2 are degenerate cases of error-correcting codes and can be used to detect single errors. The parity bit is an example of a single-error-detecting code.\n\n## Applications\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=17 \"Edit section: Applications\")\\]\n\nApplications that require low latency (such as telephone conversations) cannot use [automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Automatic_repeat_request \"Automatic repeat request\") (ARQ); they must use [forward error correction](https:\/\/en.wikipedia.org\/wiki\/Forward_error_correction \"Forward error correction\") (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be usable.\n\nApplications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available.\n\nApplications that use ARQ must have a [return channel](https:\/\/en.wikipedia.org\/wiki\/Return_channel \"Return channel\"); applications having no return channel cannot use ARQ. Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC.\n\nReliability and inspection engineering also make use of the theory of error-correcting codes,[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-16) as well as natural language.[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-DOI10.7275\/bjvb-2n37-17)\n\n### Internet\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=18 \"Edit section: Internet\")\\]\n\nIn a typical [TCP\/IP](https:\/\/en.wikipedia.org\/wiki\/TCP\/IP \"TCP\/IP\") stack, error control is performed at multiple levels:\n\n- Each [Ethernet frame](https:\/\/en.wikipedia.org\/wiki\/Ethernet_frame \"Ethernet frame\") uses [CRC-32](https:\/\/en.wikipedia.org\/wiki\/CRC-32 \"CRC-32\") error detection. Frames with detected errors are discarded by the receiver hardware.\n- The [IPv4](https:\/\/en.wikipedia.org\/wiki\/IPv4 \"IPv4\") header contains a [checksum](https:\/\/en.wikipedia.org\/wiki\/IPv4_header_checksum \"IPv4 header checksum\") protecting the contents of the header. [Packets](https:\/\/en.wikipedia.org\/wiki\/Network_packet \"Network packet\") with incorrect checksums are dropped within the network or at the receiver.\n- The checksum was omitted from the [IPv6](https:\/\/en.wikipedia.org\/wiki\/IPv6 \"IPv6\") header in order to minimize processing costs in [network routing](https:\/\/en.wikipedia.org\/wiki\/Network_routing \"Network routing\") and because current [link layer](https:\/\/en.wikipedia.org\/wiki\/Link_layer \"Link layer\") technology is assumed to provide sufficient error detection (see also RFC 3819).\n- [UDP](https:\/\/en.wikipedia.org\/wiki\/User_Datagram_Protocol \"User Datagram Protocol\") has an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the [network stack](https:\/\/en.wikipedia.org\/wiki\/Network_stack \"Network stack\"). The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection.\n- [TCP](https:\/\/en.wikipedia.org\/wiki\/Transmission_Control_Protocol \"Transmission Control Protocol\") provides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack and eventually get retransmitted using ARQ, either explicitly (such as through [three-way handshake](https:\/\/en.wikipedia.org\/wiki\/Three-way_handshake \"Three-way handshake\")) or implicitly due to a [timeout](https:\/\/en.wikipedia.org\/wiki\/Timeout_\\(computing\\) \"Timeout (computing)\").\n\n### Deep-space telecommunications\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=19 \"Edit section: Deep-space telecommunications\")\\]\n\nThe development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded) [convolutional codes](https:\/\/en.wikipedia.org\/wiki\/Convolutional_code \"Convolutional code\") and [Reed–Muller codes](https:\/\/en.wikipedia.org\/wiki\/Reed%E2%80%93Muller_code \"Reed–Muller code\").[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-deep-space-codes-18) The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a [bell curve](https:\/\/en.wikipedia.org\/wiki\/Gaussian_function \"Gaussian function\")), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977.\n\nThe [Voyager 1](https:\/\/en.wikipedia.org\/wiki\/Voyager_1 \"Voyager 1\") and [Voyager 2](https:\/\/en.wikipedia.org\/wiki\/Voyager_2 \"Voyager 2\") missions, which started in 1977, were designed to deliver color imaging and scientific information from [Jupiter](https:\/\/en.wikipedia.org\/wiki\/Jupiter \"Jupiter\") and [Saturn](https:\/\/en.wikipedia.org\/wiki\/Saturn \"Saturn\").[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-voyager-19) This resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally [Viterbi-decoded](https:\/\/en.wikipedia.org\/wiki\/Viterbi_decoder \"Viterbi decoder\")) convolutional codes that could be [concatenated](https:\/\/en.wikipedia.org\/wiki\/Concatenated_code \"Concatenated code\") with an outer [Golay (24,12,8) code](https:\/\/en.wikipedia.org\/wiki\/Binary_Golay_code \"Binary Golay code\"). The Voyager 2 craft additionally supported an implementation of a [Reed–Solomon code](https:\/\/en.wikipedia.org\/wiki\/Reed%E2%80%93Solomon_code \"Reed–Solomon code\"). The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft's extended journey to [Uranus](https:\/\/en.wikipedia.org\/wiki\/Uranus \"Uranus\") and [Neptune](https:\/\/en.wikipedia.org\/wiki\/Neptune \"Neptune\"). After ECC system upgrades in 1989, both crafts used V2 RSV coding.\n\nThe [Consultative Committee for Space Data Systems](https:\/\/en.wikipedia.org\/wiki\/Consultative_Committee_for_Space_Data_Systems \"Consultative Committee for Space Data Systems\") currently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as [Turbo codes](https:\/\/en.wikipedia.org\/wiki\/Turbo_code \"Turbo code\") or [LDPC codes](https:\/\/en.wikipedia.org\/wiki\/LDPC_code \"LDPC code\").\n\nThe different kinds of deep space and orbital missions that are conducted suggest that trying to find a one-size-fits-all error correction system will be an ongoing problem. For missions close to Earth, the nature of the [noise](https:\/\/en.wikipedia.org\/wiki\/Noise_\\(electronics\\) \"Noise (electronics)\") in the [communication channel](https:\/\/en.wikipedia.org\/wiki\/Communication_channel \"Communication channel\") is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult.\n\n### Satellite broadcasting\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=20 \"Edit section: Satellite broadcasting\")\\]\n\nThe demand for satellite [transponder](https:\/\/en.wikipedia.org\/wiki\/Transponder \"Transponder\") bandwidth continues to grow, fueled by the desire to deliver television (including new channels and [high-definition television](https:\/\/en.wikipedia.org\/wiki\/High-definition_television \"High-definition television\")) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected [modulation](https:\/\/en.wikipedia.org\/wiki\/Modulation \"Modulation\") scheme and the proportion of capacity consumed by FEC.\n\n### Data storage\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=21 \"Edit section: Data storage\")\\]\n\nError detection and correction codes are often used to improve the reliability of data storage media.[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-20) A parity track capable of detecting single-bit errors was present on the first [magnetic tape data storage](https:\/\/en.wikipedia.org\/wiki\/Magnetic_tape_data_storage \"Magnetic tape data storage\") in 1951. The [optimal rectangular code](https:\/\/en.wikipedia.org\/wiki\/Optimal_rectangular_code \"Optimal rectangular code\") used in [group coded recording](https:\/\/en.wikipedia.org\/wiki\/Group_coded_recording \"Group coded recording\") tapes not only detects but also corrects single-bit errors. Some [file formats](https:\/\/en.wikipedia.org\/wiki\/File_format \"File format\"), particularly [archive formats](https:\/\/en.wikipedia.org\/wiki\/Archive_formats \"Archive formats\"), include a checksum (most often CRC-32\\]) to detect corruption and truncation and can employ redundancy or [parity files](https:\/\/en.wikipedia.org\/wiki\/Parity_file \"Parity file\") to recover portions of corrupted data. [Reed-Solomon codes](https:\/\/en.wikipedia.org\/wiki\/Reed-Solomon_code \"Reed-Solomon code\") are used in [compact discs](https:\/\/en.wikipedia.org\/wiki\/Compact_disc \"Compact disc\") to correct errors caused by scratches.\n\nModern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors.[\\[21\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-21) [RAID](https:\/\/en.wikipedia.org\/wiki\/RAID \"RAID\") systems use a variety of error correction techniques to recover data when a hard drive completely fails. Filesystems such as [ZFS](https:\/\/en.wikipedia.org\/wiki\/ZFS \"ZFS\") or [Btrfs](https:\/\/en.wikipedia.org\/wiki\/Btrfs \"Btrfs\"), as well as some [RAID](https:\/\/en.wikipedia.org\/wiki\/RAID \"RAID\") implementations, support [data scrubbing](https:\/\/en.wikipedia.org\/wiki\/Data_scrubbing \"Data scrubbing\") and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used.[\\[22\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-22) The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware.\n\n### Error-correcting memory\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=22 \"Edit section: Error-correcting memory\")\\]\n\nMain article: [ECC memory](https:\/\/en.wikipedia.org\/wiki\/ECC_memory \"ECC memory\")\n\n[Dynamic random-access memory](https:\/\/en.wikipedia.org\/wiki\/Dynamic_random-access_memory \"Dynamic random-access memory\") (DRAM) may provide stronger protection against [soft errors](https:\/\/en.wikipedia.org\/wiki\/Soft_error \"Soft error\") by relying on error-correcting codes. Such error-correcting memory, known as *ECC* or *EDAC-protected* memory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased [radiation](https:\/\/en.wikipedia.org\/wiki\/Cosmic_ray \"Cosmic ray\") in space.\n\nError-correcting memory controllers traditionally use [Hamming codes](https:\/\/en.wikipedia.org\/wiki\/Hamming_code \"Hamming code\"), although some use [triple modular redundancy](https:\/\/en.wikipedia.org\/wiki\/Triple_modular_redundancy \"Triple modular redundancy\"). [Interleaving](https:\/\/en.wikipedia.org\/wiki\/Error_correction_code#Interleaving \"Error correction code\") allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a [single-event upset](https:\/\/en.wikipedia.org\/wiki\/Single-event_upset \"Single-event upset\") (SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained.[\\[23\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-23)\n\nIn addition to hardware providing features required for ECC memory to operate, [operating systems](https:\/\/en.wikipedia.org\/wiki\/Operating_system \"Operating system\") usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the [Linux kernel](https:\/\/en.wikipedia.org\/wiki\/Linux_kernel \"Linux kernel\")'s *EDAC* subsystem (previously known as *Bluesmoke*), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the [PCI bus](https:\/\/en.wikipedia.org\/wiki\/PCI_bus \"PCI bus\").[\\[24\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-24)[\\[25\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-25)[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-26) A few systems\\[*[specify](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citing_sources \"Wikipedia:Citing sources\")*\\] also support [memory scrubbing](https:\/\/en.wikipedia.org\/wiki\/Memory_scrubbing \"Memory scrubbing\") to catch and correct errors early before they become unrecoverable.\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=23 \"Edit section: See also\")\\]\n\n- [Berger code](https:\/\/en.wikipedia.org\/wiki\/Berger_code \"Berger code\")\n- [Burst error-correcting code](https:\/\/en.wikipedia.org\/wiki\/Burst_error-correcting_code \"Burst error-correcting code\")\n- [ECC memory](https:\/\/en.wikipedia.org\/wiki\/ECC_memory \"ECC memory\"), a type of computer data storage\n- [Link adaptation](https:\/\/en.wikipedia.org\/wiki\/Link_adaptation \"Link adaptation\")\n- [List of algorithms § Error detection and correction](https:\/\/en.wikipedia.org\/wiki\/List_of_algorithms#Error_detection_and_correction \"List of algorithms\")\n- [List of hash functions](https:\/\/en.wikipedia.org\/wiki\/List_of_hash_functions \"List of hash functions\")\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=24 \"Edit section: References\")\\]\n\n1. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-Jewish_1-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-Jewish_1-1)\n   \"Masorah\". [*Jewish Encyclopedia*](http:\/\/www.jewishencyclopedia.com\/articles\/10465-masorah).\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-2)**\n   Pratico, Gary D.; Pelt, Miles V. Van (2009). [*Basics of Biblical Hebrew Grammar: Second Edition*](https:\/\/books.google.com\/books?id=p3FIoT0s3yYC&pg=PT448). Zondervan. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-310-55882-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-310-55882-8 \"Special:BookSources\/978-0-310-55882-8\")\n   \n   .\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-3)**\n   Mounce, William D. (2007). [*Greek for the Rest of Us: Using Greek Tools Without Mastering Biblical Languages*](https:\/\/books.google.com\/books?id=AF-5ptJ0l2gC&pg=PA289). Zondervan. p. 289. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-310-28289-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-310-28289-1 \"Special:BookSources\/978-0-310-28289-1\")\n   \n   .\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-4)**\n   [Mishneh Torah](https:\/\/en.wikipedia.org\/wiki\/Mishneh_Torah \"Mishneh Torah\"), Tefillin, Mezuzah, and Sefer Torah, 1:2. Example English translation:\n   \n   [Eliyahu Touger](https:\/\/en.wikipedia.org\/w\/index.php?title=Eliyahu_Touger&action=edit&redlink=1 \"Eliyahu Touger (page does not exist)\"). [*The Rambam's Mishneh Torah*](https:\/\/www.chabad.org\/library\/article_cdo\/aid\/925417\/jewish\/Tefillin-Mezuzah-and-Sefer-Torah-Chapter-One.htm). [Moznaim Publishing Corporation](https:\/\/en.wikipedia.org\/w\/index.php?title=Moznaim_Publishing_Corporation&action=edit&redlink=1 \"Moznaim Publishing Corporation (page does not exist)\").\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-5)**\n   Brian M. Fagan (5 December 1996). \"Dead Sea Scrolls\". *[The Oxford Companion to Archaeology](https:\/\/en.wikipedia.org\/w\/index.php?title=The_Oxford_Companion_to_Archaeology&action=edit&redlink=1 \"The Oxford Companion to Archaeology (page does not exist)\")*. [Oxford University Press](https:\/\/en.wikipedia.org\/wiki\/Oxford_University_Press \"Oxford University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0195076184](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0195076184 \"Special:BookSources\/0195076184\")\n   \n   .\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-Thompson_6-0)**\n   Thompson, Thomas M. (1983), *From Error-Correcting Codes through Sphere Packings to Simple Groups*, The Carus Mathematical Monographs (\\#21), The Mathematical Association of America, p. vii, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-88385-023-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-88385-023-0 \"Special:BookSources\/0-88385-023-0\")\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-7)**\n   Shannon, C.E. (1948), \"A Mathematical Theory of Communication\", *Bell System Technical Journal*, **27** (3): 379–423, [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/j.1538-7305.1948.tb01338.x](https:\/\/doi.org\/10.1002%2Fj.1538-7305.1948.tb01338.x), [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10338\\.dmlcz\/101429](https:\/\/hdl.handle.net\/10338.dmlcz%2F101429), [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [9230594](https:\/\/pubmed.ncbi.nlm.nih.gov\/9230594)\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-8)**\n   Golay, Marcel J. E. (1949), \"Notes on Digital Coding\", *Proc.I.R.E. (I.E.E.E.)*, **37**: 657\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-9)**\n   Gupta, Vikas; Verma, Chanderkant (November 2012). \"Error Detection and Correction: An Introduction\". *International Journal of Advanced Research in Computer Science and Software Engineering*. **2** (11). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [17499858](https:\/\/api.semanticscholar.org\/CorpusID:17499858).\n10. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-reliable-erasure-code_10-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-reliable-erasure-code_10-1) A. J. McAuley, *Reliable Broadband Communication Using a Burst Erasure Correcting Code*, ACM SIGCOMM, 1990.\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-11)**\n    Shah, Pradeep M.; Vyavahare, Prakash D.; Jain, Anjana (September 2015). [\"Modern error correcting codes for 4G and beyond: Turbo codes and LDPC codes\"](https:\/\/www.researchgate.net\/publication\/301611980). *2015 Radio and Antenna Days of the Indian Ocean (RADIO)*. pp. 1–2\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1109\/RADIO.2015.7323369](https:\/\/doi.org\/10.1109%2FRADIO.2015.7323369). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-9-9903-7339-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-9-9903-7339-4 \"Special:BookSources\/978-9-9903-7339-4\")\n    \n    . [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [28885076](https:\/\/api.semanticscholar.org\/CorpusID:28885076). Retrieved 22 May 2022.\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-12)**\n    [\"IEEE SA - IEEE 802.11ac-2013\"](https:\/\/web.archive.org\/web\/20220522205452\/https:\/\/standards.ieee.org\/ieee\/802.11ac\/4473\/). *IEEE Standards Association*. Archived from [the original](https:\/\/standards.ieee.org\/ieee\/802.11ac\/4473\/) on 2022-05-22. Retrieved 2022-05-22.\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-13)**\n    [\"Transition to Advanced Format 4K Sector Hard Drives \\| Seagate US\"](https:\/\/www.seagate.com\/sg\/en\/tech-insights\/advanced-format-4k-sector-hard-drives-master-ti\/). *Seagate.com*. Retrieved 22 May 2022.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-14)**\n    Frank van Gerwen. [\"Numbers (and other mysterious) stations\"](https:\/\/web.archive.org\/web\/20170712053335\/http:\/\/www.cisquet.nl\/numbers.htm). Archived from [the original](http:\/\/www.cisquet.nl\/numbers.htm) on 12 July 2017. Retrieved 12 March 2012.\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-15)**\n    Gary Cutlack (25 August 2010). [\"Mysterious Russian 'Numbers Station' Changes Broadcast After 20 Years\"](https:\/\/web.archive.org\/web\/20170705131816\/https:\/\/www.gizmodo.com.au\/2010\/08\/mysterious-russian-numbers-station-changes-broadcast-after-20-years\/). *Gizmodo*. Archived from [the original](http:\/\/www.gizmodo.com.au\/2010\/08\/mysterious-russian-numbers-station-changes-broadcast-after-20-years\/) on 5 July 2017. Retrieved 12 March 2012.\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-16)**\n    Ben-Gal I.; Herer Y.; Raz T. (2003). [\"Self-correcting inspection procedure under inspection errors\"](https:\/\/web.archive.org\/web\/20131013171945\/http:\/\/www.eng.tau.ac.il\/~bengal\/SCI_paper.pdf) (PDF). *IIE Transactions*. IIE Transactions on Quality and Reliability, 34(6), pp. 529-540. Archived from [the original](http:\/\/www.eng.tau.ac.il\/~bengal\/SCI_paper.pdf) (PDF) on 2013-10-13. Retrieved 2014-01-10.\n17. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-DOI10.7275\/bjvb-2n37_17-0)**\n    Yvo Meeres, Tommi A. Pirinen (2021). \"Vowel Harmony Viewed as Error-Correcting Code\". *Proceedings of the Society for Computation in Linguistics*. **4** (1): 313–322\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.7275\/bjvb-2n37](https:\/\/doi.org\/10.7275%2Fbjvb-2n37).\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-deep-space-codes_18-0)** K. Andrews et al., *The Development of Turbo and LDPC Codes for Deep-Space Applications*, Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007.\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-voyager_19-0)**\n    Huffman, William Cary; [Pless, Vera S.](https:\/\/en.wikipedia.org\/wiki\/Vera_Pless \"Vera Pless\") (2003). [*Fundamentals of Error-Correcting Codes*](https:\/\/archive.org\/details\/fundamentalsofer0000huff). [Cambridge University Press](https:\/\/en.wikipedia.org\/wiki\/Cambridge_University_Press \"Cambridge University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-521-78280-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-521-78280-7 \"Special:BookSources\/978-0-521-78280-7\")\n    \n    .\n20. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-20)**\n    Kurtas, Erozan M.; Vasic, Bane (2018-10-03). *Advanced Error Control Techniques for Data Storage Systems*. CRC Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4200-3649-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4200-3649-7 \"Special:BookSources\/978-1-4200-3649-7\")\n    \n    .\n21. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-21)**\n    Scott A. Moulton. [\"My Hard Drive Died\"](https:\/\/web.archive.org\/web\/20080202143103\/http:\/\/www.myharddrivedied.com\/presentations_whitepaper.html). Archived from [the original](http:\/\/www.myharddrivedied.com\/presentations_whitepaper.html) on 2008-02-02.\n22. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-22)**\n    Qiao, Zhi; Fu, Song; Chen, Hsing-Bung; Settlemyer, Bradley (2019). \"Building Reliable High-Performance Storage Systems: An Empirical and Analytical Study\". *2019 IEEE International Conference on Cluster Computing (CLUSTER)*. pp. 1–10\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1109\/CLUSTER.2019.8891006](https:\/\/doi.org\/10.1109%2FCLUSTER.2019.8891006). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-7281-4734-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-7281-4734-5 \"Special:BookSources\/978-1-7281-4734-5\")\n    \n    . [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [207951690](https:\/\/api.semanticscholar.org\/CorpusID:207951690).\n23. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-23)**\n    [\"Using StrongArm SA-1110 in the On-Board Computer of Nanosatellite\"](https:\/\/web.archive.org\/web\/20111002152735\/http:\/\/www.apmcsta.org\/File\/doc\/Conferences\/6th%20meeting\/Chen%20Zhenyu.doc). Tsinghua Space Center, [Tsinghua University](https:\/\/en.wikipedia.org\/wiki\/Tsinghua_University \"Tsinghua University\"), Beijing. Archived from [the original](http:\/\/www.apmcsta.org\/File\/doc\/Conferences\/6th%20meeting\/Chen%20Zhenyu.doc) on 2011-10-02. Retrieved 2009-02-16.\n24. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-24)**\n    Jeff Layton. [\"Error Detection and Correction\"](http:\/\/www.admin-magazine.com\/Articles\/Monitoring-Memory-Errors). *[Linux Magazine](https:\/\/en.wikipedia.org\/wiki\/Linux_Magazine \"Linux Magazine\")*. Retrieved 2014-08-12.\n25. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-25)**\n    [\"EDAC Project\"](https:\/\/bluesmoke.sourceforge.net\/). *bluesmoke.sourceforge.net*. Retrieved 2014-08-12.\n26. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-26)**\n    [\"Documentation\/edac.txt\"](https:\/\/web.archive.org\/web\/20090905174616\/http:\/\/www.kernel.org\/doc\/Documentation\/edac.txt). *Linux kernel documentation*. [kernel.org](https:\/\/en.wikipedia.org\/wiki\/Kernel.org \"Kernel.org\"). 2014-06-16. Archived from [the original](https:\/\/www.kernel.org\/doc\/Documentation\/edac.txt) on 2009-09-05. Retrieved 2014-08-12.\n\n## Further reading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=25 \"Edit section: Further reading\")\\]\n\n- Shu Lin; Daniel J. Costello, Jr. (1983). *Error Control Coding: Fundamentals and Applications*. [Prentice Hall](https:\/\/en.wikipedia.org\/wiki\/Prentice_Hall \"Prentice Hall\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [0-13-283796-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-13-283796-X \"Special:BookSources\/0-13-283796-X\")\n  \n  .\n- [SoftECC: A System for Software Memory Integrity Checking](http:\/\/pdos.csail.mit.edu\/papers\/softecc:ddopson-meng\/softecc_ddopson-meng.pdf)\n- [A Tunable, Software-based DRAM Error Detection and Correction Library for HPC](http:\/\/www.fiala.me\/pubs\/papers\/libsdc11.pdf) [Archived](https:\/\/web.archive.org\/web\/20141107074502\/http:\/\/www.fiala.me\/pubs\/papers\/libsdc11.pdf) 2014-11-07 at the [Wayback Machine](https:\/\/en.wikipedia.org\/wiki\/Wayback_Machine \"Wayback Machine\")\n- [Detection and Correction of Silent Data Corruption for Large-Scale High-Performance Computing](http:\/\/www.fiala.me\/pubs\/papers\/sc12-redmpi.pdf) [Archived](https:\/\/web.archive.org\/web\/20141107074511\/http:\/\/www.fiala.me\/pubs\/papers\/sc12-redmpi.pdf) 2014-11-07 at the [Wayback Machine](https:\/\/en.wikipedia.org\/wiki\/Wayback_Machine \"Wayback Machine\")\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=26 \"Edit section: External links\")\\]\n\n- [The on-line textbook: Information Theory, Inference, and Learning Algorithms](http:\/\/www.inference.phy.cam.ac.uk\/mackay\/itila\/), by [David J.C. MacKay](https:\/\/en.wikipedia.org\/wiki\/David_J.C._MacKay \"David J.C. MacKay\"), contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including [low-density parity-check codes](https:\/\/en.wikipedia.org\/wiki\/Low-density_parity-check_code \"Low-density parity-check code\"), [turbo codes](https:\/\/en.wikipedia.org\/wiki\/Turbo_code \"Turbo code\"), and [fountain codes](https:\/\/en.wikipedia.org\/wiki\/Fountain_codes \"Fountain codes\").\n- [ECC Page](http:\/\/www.eccpage.com\/) – implementations of popular ECC encoding and decoding routines\n\n![](https:\/\/en.wikipedia.org\/wiki\/Special:CentralAutoLogin\/start?useformat=desktop&type=1x1&usesul3=1)\n\nRetrieved from \"<https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&oldid=1337702567>\"\n\n[Categories](https:\/\/en.wikipedia.org\/wiki\/Help:Category \"Help:Category\"):\n\n- [Error detection and correction](https:\/\/en.wikipedia.org\/wiki\/Category:Error_detection_and_correction \"Category:Error detection and correction\")\n- [Computer errors](https:\/\/en.wikipedia.org\/wiki\/Category:Computer_errors \"Category:Computer errors\")\n\nHidden categories:\n\n- [Articles with short description](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_short_description \"Category:Articles with short description\")\n- [Short description is different from Wikidata](https:\/\/en.wikipedia.org\/wiki\/Category:Short_description_is_different_from_Wikidata \"Category:Short description is different from Wikidata\")\n- [Articles needing additional references from August 2008](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_needing_additional_references_from_August_2008 \"Category:Articles needing additional references from August 2008\")\n- [All articles needing additional references](https:\/\/en.wikipedia.org\/wiki\/Category:All_articles_needing_additional_references \"Category:All articles needing additional references\")\n- [Articles needing more detailed references](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_needing_more_detailed_references \"Category:Articles needing more detailed references\")\n- [Webarchive template wayback links](https:\/\/en.wikipedia.org\/wiki\/Category:Webarchive_template_wayback_links \"Category:Webarchive template wayback links\")\n\n- This page was last edited on 11 February 2026, at 00:30 (UTC).\n- Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License \"Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License\"); additional terms may apply. By using this site, you agree to the [Terms of Use](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Terms_of_Use \"foundation:Special:MyLanguage\/Policy:Terms of Use\") and [Privacy Policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy \"foundation:Special:MyLanguage\/Policy:Privacy policy\"). Wikipedia® is a registered trademark of the [Wikimedia Foundation, Inc.](https:\/\/wikimediafoundation.org\/), a non-profit organization.\n\n- [Privacy policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy)\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About)\n- [Disclaimers](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:General_disclaimer)\n- [Contact Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us)\n- [Legal & safety contacts](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Legal:Wikimedia_Foundation_Legal_and_Safety_Contact_Information)\n- [Code of Conduct](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Universal_Code_of_Conduct)\n- [Developers](https:\/\/developer.wikimedia.org\/)\n- [Statistics](https:\/\/stats.wikimedia.org\/#\/en.wikipedia.org)\n- [Cookie statement](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Cookie_statement)\n- [Mobile view](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&mobileaction=toggle_view_mobile)\n\n- [![Wikimedia Foundation](https:\/\/en.wikipedia.org\/static\/images\/footer\/wikimedia.svg)](https:\/\/www.wikimedia.org\/)\n- [![Powered by MediaWiki](https:\/\/en.wikipedia.org\/w\/resources\/assets\/mediawiki_compact.svg)](https:\/\/www.mediawiki.org\/)\n\nSearch\n\nToggle the table of contents\n\nError detection and correction\n\n36 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction)","attrs_readable_markdown":"[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/ef\/Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg\/250px-Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg)\n\nTo clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied [Reed–Solomon error correction](https:\/\/en.wikipedia.org\/wiki\/Reed%E2%80%93Solomon_error_correction \"Reed–Solomon error correction\") (right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted.\n\nIn [information theory](https:\/\/en.wikipedia.org\/wiki\/Information_theory \"Information theory\") and [coding theory](https:\/\/en.wikipedia.org\/wiki\/Coding_theory \"Coding theory\") with applications in [computer science](https:\/\/en.wikipedia.org\/wiki\/Computer_science \"Computer science\") and [telecommunications](https:\/\/en.wikipedia.org\/wiki\/Telecommunications \"Telecommunications\"), **error detection and correction** (**EDAC**) or **error control** are techniques that enable [reliable delivery](https:\/\/en.wikipedia.org\/wiki\/Reliable_delivery \"Reliable delivery\") of [digital data](https:\/\/en.wikipedia.org\/wiki\/Digital_data \"Digital data\") over unreliable [communication channels](https:\/\/en.wikipedia.org\/wiki\/Communication_channel \"Communication channel\"). Many communication channels are subject to [channel noise](https:\/\/en.wikipedia.org\/wiki\/Channel_noise \"Channel noise\"), and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases.\n\n*Error detection* is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver.\n\n*Error correction* is the detection of errors and reconstruction of the original, error-free data.\n\nIn classical antiquity, [copyists](https:\/\/en.wikipedia.org\/wiki\/Copyist \"Copyist\") of the [Hebrew Bible](https:\/\/en.wikipedia.org\/wiki\/Hebrew_Bible \"Hebrew Bible\") were paid for their work according to the number of [stichs](https:\/\/en.wiktionary.org\/wiki\/stich \"wiktionary:stich\") (lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-Jewish-1) This also helped ensure accuracy in the transmission of the text with the production of subsequent copies.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-2)[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-3) Between the 7th and 10th centuries CE a [group of Jewish scribes](https:\/\/en.wikipedia.org\/wiki\/Masoretes \"Masoretes\") formalized and expanded this to create the [Numerical Masorah](https:\/\/en.wikipedia.org\/wiki\/Masoretic_Text#Numerical_Masorah \"Masoretic Text\") to ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-Jewish-1) Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-4) The effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the [Dead Sea Scrolls](https:\/\/en.wikipedia.org\/wiki\/Dead_Sea_Scrolls \"Dead Sea Scrolls\") in 1947–1956, dating from c. 150 BCE – 75 CE.[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-5)\n\nThe modern development of [error correction codes](https:\/\/en.wikipedia.org\/wiki\/Error_correction_code \"Error correction code\") is credited to [Richard Hamming](https:\/\/en.wikipedia.org\/wiki\/Richard_Hamming \"Richard Hamming\") in 1947.[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-Thompson-6) A description of [Hamming's code](https:\/\/en.wikipedia.org\/wiki\/Hamming%27s_code \"Hamming's code\") appeared in [Claude Shannon](https:\/\/en.wikipedia.org\/wiki\/Claude_Shannon \"Claude Shannon\")'s *A Mathematical Theory of Communication*[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-7) and was quickly generalized by [Marcel J. E. Golay](https:\/\/en.wikipedia.org\/wiki\/Marcel_J._E._Golay \"Marcel J. E. Golay\").[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-8)\n\nAll error-detection and correction schemes add some [redundancy](https:\/\/en.wikipedia.org\/wiki\/Redundancy_\\(information_theory\\) \"Redundancy (information theory)\") (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted. Error detection and correction schemes can be either [systematic](https:\/\/en.wikipedia.org\/wiki\/Systematic_code \"Systematic code\") or non-systematic. In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of *check bits* (or *parity data*), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message.\n\nGood error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common [channel models](https:\/\/en.wikipedia.org\/wiki\/Channel_model \"Channel model\") include [memoryless](https:\/\/en.wikipedia.org\/wiki\/Memoryless \"Memoryless\") models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in [bursts](https:\/\/en.wikipedia.org\/wiki\/Burst_error \"Burst error\"). Consequently, error-detecting and -correcting codes can be generally distinguished between *random-error-detecting\/correcting* and *burst-error-detecting\/correcting*. Some codes can also be suitable for a mixture of random errors and burst errors.\n\nIf the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as [automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Automatic_repeat_request \"Automatic repeat request\") (ARQ), and is most notably used in the Internet. An alternate approach for error control is [hybrid automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Hybrid_automatic_repeat_request \"Hybrid automatic repeat request\") (HARQ), which is a combination of ARQ and error-correction coding.\n\n## Types of error correction\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=4 \"Edit section: Types of error correction\")\\]\n\nThere are three major types of error correction:[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-9)\n\n### Automatic repeat request\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=5 \"Edit section: Automatic repeat request\")\\]\n\n[Automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Automatic_repeat_request \"Automatic repeat request\") (ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and\/or negative acknowledgment messages, and [timeouts](https:\/\/en.wikipedia.org\/wiki\/Timeout_\\(computing\\) \"Timeout (computing)\") to achieve reliable data transmission. An *acknowledgment* is a message sent by the receiver to indicate that it has correctly received a [data frame](https:\/\/en.wikipedia.org\/wiki\/Data_frame \"Data frame\").\n\nUsually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e., within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions.\n\nThree types of ARQ protocols are [Stop-and-wait ARQ](https:\/\/en.wikipedia.org\/wiki\/Stop-and-wait_ARQ \"Stop-and-wait ARQ\"), [Go-Back-N ARQ](https:\/\/en.wikipedia.org\/wiki\/Go-Back-N_ARQ \"Go-Back-N ARQ\"), and [Selective Repeat ARQ](https:\/\/en.wikipedia.org\/wiki\/Selective_Repeat_ARQ \"Selective Repeat ARQ\").\n\nARQ is appropriate if the communication channel has varying or unknown [capacity](https:\/\/en.wikipedia.org\/wiki\/Channel_capacity \"Channel capacity\"), such as is the case on the Internet. However, ARQ requires the availability of a [back channel](https:\/\/en.wikipedia.org\/wiki\/Backward_channel \"Backward channel\"), results in possibly increased [latency](https:\/\/en.wikipedia.org\/wiki\/Network_latency \"Network latency\") due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of [network congestion](https:\/\/en.wikipedia.org\/wiki\/Network_congestion \"Network congestion\") can put a strain on the server and overall network capacity.[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-reliable-erasure-code-10)\n\nFor example, ARQ is used on shortwave radio data links in the form of [ARQ-E](https:\/\/en.wikipedia.org\/wiki\/ARQ-E \"ARQ-E\"), or combined with multiplexing as [ARQ-M](https:\/\/en.wikipedia.org\/wiki\/ARQ-M \"ARQ-M\").\n\n### Forward error correction\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=6 \"Edit section: Forward error correction\")\\]\n\n[Forward error correction](https:\/\/en.wikipedia.org\/wiki\/Forward_error_correction \"Forward error correction\") (FEC) is a process of adding [redundant data](https:\/\/en.wikipedia.org\/wiki\/Redundancy_\\(information_theory\\) \"Redundancy (information theory)\") such as an [error-correcting code](https:\/\/en.wikipedia.org\/wiki\/Error-correcting_code \"Error-correcting code\") (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a [backchannel](https:\/\/en.wikipedia.org\/wiki\/Backchannel \"Backchannel\") is not required in forward error correction. Error-correcting codes are used in [lower-layer](https:\/\/en.wikipedia.org\/wiki\/Physical_layer \"Physical layer\") communication such as [cellular network](https:\/\/en.wikipedia.org\/wiki\/Cellular_network \"Cellular network\"), high-speed [fiber-optic communication](https:\/\/en.wikipedia.org\/wiki\/Fiber-optic_communication \"Fiber-optic communication\") and [Wi-Fi](https:\/\/en.wikipedia.org\/wiki\/Wi-Fi \"Wi-Fi\"),[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-11)[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-12) as well as for reliable storage in media such as [flash memory](https:\/\/en.wikipedia.org\/wiki\/Flash_memory \"Flash memory\"), [hard disk](https:\/\/en.wikipedia.org\/wiki\/Hard_disk \"Hard disk\") and [RAM](https:\/\/en.wikipedia.org\/wiki\/ECC_memory \"ECC memory\").[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-13)\n\nError-correcting codes are usually distinguished between [convolutional codes](https:\/\/en.wikipedia.org\/wiki\/Convolutional_code \"Convolutional code\") and [block codes](https:\/\/en.wikipedia.org\/wiki\/Block_code \"Block code\"):\n\n- *Convolutional codes* are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the [Viterbi decoder](https:\/\/en.wikipedia.org\/wiki\/Viterbi_decoder \"Viterbi decoder\") allows [optimal decoding](https:\/\/en.wikipedia.org\/wiki\/Maximum_likelihood_decoding \"Maximum likelihood decoding\").\n- *Block codes* are processed on a [block-by-block](https:\/\/en.wikipedia.org\/wiki\/Block_\\(telecommunications\\) \"Block (telecommunications)\") basis. Early examples of block codes are [repetition codes](https:\/\/en.wikipedia.org\/wiki\/Repetition_code \"Repetition code\"), [Hamming codes](https:\/\/en.wikipedia.org\/wiki\/Hamming_code \"Hamming code\") and [multidimensional parity-check codes](https:\/\/en.wikipedia.org\/wiki\/Multidimensional_parity-check_code \"Multidimensional parity-check code\"). They were followed by a number of efficient codes, [Reed–Solomon codes](https:\/\/en.wikipedia.org\/wiki\/Reed%E2%80%93Solomon_code \"Reed–Solomon code\") being the most notable due to their current widespread use. [Turbo codes](https:\/\/en.wikipedia.org\/wiki\/Turbo_code \"Turbo code\") and [low-density parity-check codes](https:\/\/en.wikipedia.org\/wiki\/Low-density_parity-check_code \"Low-density parity-check code\") (LDPC) are relatively new constructions that can provide almost [optimal efficiency](https:\/\/en.wikipedia.org\/wiki\/Category:Capacity-approaching_codes \"Category:Capacity-approaching codes\").\n\n[Shannon's theorem](https:\/\/en.wikipedia.org\/wiki\/Shannon%27s_theorem \"Shannon's theorem\") is an important theorem in forward error correction, and describes the maximum [information rate](https:\/\/en.wikipedia.org\/wiki\/Information_rate \"Information rate\") at which reliable communication is possible over a channel that has a certain error probability or [signal-to-noise ratio](https:\/\/en.wikipedia.org\/wiki\/Signal-to-noise_ratio \"Signal-to-noise ratio\") (SNR). This strict upper limit is expressed in terms of the [channel capacity](https:\/\/en.wikipedia.org\/wiki\/Channel_capacity \"Channel capacity\"). More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a [discrete memoryless channel](https:\/\/en.wikipedia.org\/wiki\/Channel_model \"Channel model\") can be made arbitrarily small, provided that the [code rate](https:\/\/en.wikipedia.org\/wiki\/Code_rate \"Code rate\") is smaller than the channel capacity. The code rate is defined as the fraction *k\/n* of *k* source symbols and *n* encoded symbols.\n\nThe actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have [efficient](https:\/\/en.wikipedia.org\/wiki\/Polynomial_time \"Polynomial time\") encoding and decoding algorithms.\n\n[Hybrid ARQ](https:\/\/en.wikipedia.org\/wiki\/Hybrid_ARQ \"Hybrid ARQ\") is a combination of ARQ and forward error correction. There are two basic approaches:[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-reliable-erasure-code-10)\n\n- Messages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check).\n- Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message.\n\nThe latter approach is particularly attractive on an [erasure channel](https:\/\/en.wikipedia.org\/wiki\/Erasure_channel \"Erasure channel\") when using a [rateless erasure code](https:\/\/en.wikipedia.org\/wiki\/Rateless_erasure_code \"Rateless erasure code\").\n\n## Types of error detection\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=8 \"Edit section: Types of error detection\")\\]\n\nError detection is most commonly realized using a suitable [hash function](https:\/\/en.wikipedia.org\/wiki\/Hash_function \"Hash function\") (or specifically, a [checksum](https:\/\/en.wikipedia.org\/wiki\/Checksum \"Checksum\"), [cyclic redundancy check](https:\/\/en.wikipedia.org\/wiki\/Cyclic_redundancy_check \"Cyclic redundancy check\") or other algorithm). A hash function adds a fixed-length *tag* to a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided.\n\nThere exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check's performance in detecting [burst errors](https:\/\/en.wikipedia.org\/wiki\/Burst_error \"Burst error\")).\n\n### Minimum distance coding\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=9 \"Edit section: Minimum distance coding\")\\]\n\nA random-error-correcting code based on [minimum distance coding](https:\/\/en.wikipedia.org\/wiki\/Minimum_distance_coding \"Minimum distance coding\") can provide a strict guarantee on the number of detectable errors, but it may not protect against a [preimage attack](https:\/\/en.wikipedia.org\/wiki\/Preimage_attack \"Preimage attack\").\n\nA [repetition code](https:\/\/en.wikipedia.org\/wiki\/Repetition_code \"Repetition code\") is a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern 1011, the four-bit block can be repeated three times, thus producing 1011 1011 1011. If this twelve-bit pattern was received as 1010 1011 1011 – where the first block is unlike the other two – an error has occurred.\n\nA repetition code is very inefficient and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g., 1010 1010 1010 in the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of [numbers stations](https:\/\/en.wikipedia.org\/wiki\/Numbers_station \"Numbers station\").[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-14)[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-15)\n\nA *parity bit* is a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous.\n\nParity bits added to each *word* sent are called [transverse redundancy checks](https:\/\/en.wikipedia.org\/wiki\/Transverse_redundancy_check \"Transverse redundancy check\"), while those added at the end of a stream of *words* are called [longitudinal redundancy checks](https:\/\/en.wikipedia.org\/wiki\/Longitudinal_redundancy_check \"Longitudinal redundancy check\"). For example, if each of a series of m-bit *words* has a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words. With more error correction bits, more errors can be detected and in some cases corrected.\n\nThere are also other bit-grouping techniques.\n\nA *checksum* of a message is a [modular arithmetic](https:\/\/en.wikipedia.org\/wiki\/Modular_arithmetic \"Modular arithmetic\") sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a [ones'-complement](https:\/\/en.wikipedia.org\/wiki\/Ones%27-complement \"Ones'-complement\") operation prior to transmission to detect unintentional all-zero messages.\n\nChecksum schemes include parity bits, [check digits](https:\/\/en.wikipedia.org\/wiki\/Check_digit \"Check digit\"), and [longitudinal redundancy checks](https:\/\/en.wikipedia.org\/wiki\/Longitudinal_redundancy_check \"Longitudinal redundancy check\"). Some checksum schemes, such as the [Damm algorithm](https:\/\/en.wikipedia.org\/wiki\/Damm_algorithm \"Damm algorithm\"), the [Luhn algorithm](https:\/\/en.wikipedia.org\/wiki\/Luhn_algorithm \"Luhn algorithm\"), and the [Verhoeff algorithm](https:\/\/en.wikipedia.org\/wiki\/Verhoeff_algorithm \"Verhoeff algorithm\"), are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers.\n\n### Cyclic redundancy check\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=13 \"Edit section: Cyclic redundancy check\")\\]\n\nA *cyclic redundancy check* (CRC) is a non-secure [hash function](https:\/\/en.wikipedia.org\/wiki\/Hash_function \"Hash function\") designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a *generator polynomial*, which is used as the [divisor](https:\/\/en.wikipedia.org\/wiki\/Divisor \"Divisor\") in a [polynomial long division](https:\/\/en.wikipedia.org\/wiki\/Polynomial_long_division \"Polynomial long division\") over a [finite field](https:\/\/en.wikipedia.org\/wiki\/Finite_field \"Finite field\"), taking the input data as the [dividend](https:\/\/en.wikipedia.org\/wiki\/Dividend \"Dividend\"). The [remainder](https:\/\/en.wikipedia.org\/wiki\/Remainder \"Remainder\") becomes the result.\n\nA CRC has properties that make it well suited for detecting [burst errors](https:\/\/en.wikipedia.org\/wiki\/Burst_error \"Burst error\"). CRCs are particularly easy to implement in hardware and are therefore commonly used in [computer networks](https:\/\/en.wikipedia.org\/wiki\/Computer_network \"Computer network\") and storage devices such as [hard disk drives](https:\/\/en.wikipedia.org\/wiki\/Hard_disk_drives \"Hard disk drives\").\n\nThe parity bit can be seen as a special-case 1-bit CRC.\n\n### Cryptographic hash function\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=14 \"Edit section: Cryptographic hash function\")\\]\n\nThe output of a *cryptographic hash function*, also known as a *message digest*, can provide strong assurances about [data integrity](https:\/\/en.wikipedia.org\/wiki\/Data_integrity \"Data integrity\"), whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a *keyed hash* or [message authentication code](https:\/\/en.wikipedia.org\/wiki\/Message_authentication_code \"Message authentication code\") (MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message.\n\nDigital signatures can provide strong assurances about data integrity, whether the changes of the data are accidental or maliciously introduced. Digital signatures are perhaps most notable for being part of the HTTPS protocol for securely browsing the web.\n\n### Error correction code\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=16 \"Edit section: Error correction code\")\\]\n\nAny error-correcting code can be used for error detection. A code with *minimum [Hamming distance](https:\/\/en.wikipedia.org\/wiki\/Hamming_distance \"Hamming distance\")*, *d*, can detect up to *d* − 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired.\n\nCodes with minimum Hamming distance *d* = 2 are degenerate cases of error-correcting codes and can be used to detect single errors. The parity bit is an example of a single-error-detecting code.\n\nApplications that require low latency (such as telephone conversations) cannot use [automatic repeat request](https:\/\/en.wikipedia.org\/wiki\/Automatic_repeat_request \"Automatic repeat request\") (ARQ); they must use [forward error correction](https:\/\/en.wikipedia.org\/wiki\/Forward_error_correction \"Forward error correction\") (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be usable.\n\nApplications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available.\n\nApplications that use ARQ must have a [return channel](https:\/\/en.wikipedia.org\/wiki\/Return_channel \"Return channel\"); applications having no return channel cannot use ARQ. Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC.\n\nReliability and inspection engineering also make use of the theory of error-correcting codes,[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-16) as well as natural language.[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-DOI10.7275\/bjvb-2n37-17)\n\nIn a typical [TCP\/IP](https:\/\/en.wikipedia.org\/wiki\/TCP\/IP \"TCP\/IP\") stack, error control is performed at multiple levels:\n\n- Each [Ethernet frame](https:\/\/en.wikipedia.org\/wiki\/Ethernet_frame \"Ethernet frame\") uses [CRC-32](https:\/\/en.wikipedia.org\/wiki\/CRC-32 \"CRC-32\") error detection. Frames with detected errors are discarded by the receiver hardware.\n- The [IPv4](https:\/\/en.wikipedia.org\/wiki\/IPv4 \"IPv4\") header contains a [checksum](https:\/\/en.wikipedia.org\/wiki\/IPv4_header_checksum \"IPv4 header checksum\") protecting the contents of the header. [Packets](https:\/\/en.wikipedia.org\/wiki\/Network_packet \"Network packet\") with incorrect checksums are dropped within the network or at the receiver.\n- The checksum was omitted from the [IPv6](https:\/\/en.wikipedia.org\/wiki\/IPv6 \"IPv6\") header in order to minimize processing costs in [network routing](https:\/\/en.wikipedia.org\/wiki\/Network_routing \"Network routing\") and because current [link layer](https:\/\/en.wikipedia.org\/wiki\/Link_layer \"Link layer\") technology is assumed to provide sufficient error detection (see also RFC 3819).\n- [UDP](https:\/\/en.wikipedia.org\/wiki\/User_Datagram_Protocol \"User Datagram Protocol\") has an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the [network stack](https:\/\/en.wikipedia.org\/wiki\/Network_stack \"Network stack\"). The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection.\n- [TCP](https:\/\/en.wikipedia.org\/wiki\/Transmission_Control_Protocol \"Transmission Control Protocol\") provides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack and eventually get retransmitted using ARQ, either explicitly (such as through [three-way handshake](https:\/\/en.wikipedia.org\/wiki\/Three-way_handshake \"Three-way handshake\")) or implicitly due to a [timeout](https:\/\/en.wikipedia.org\/wiki\/Timeout_\\(computing\\) \"Timeout (computing)\").\n\n### Deep-space telecommunications\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=19 \"Edit section: Deep-space telecommunications\")\\]\n\nThe development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded) [convolutional codes](https:\/\/en.wikipedia.org\/wiki\/Convolutional_code \"Convolutional code\") and [Reed–Muller codes](https:\/\/en.wikipedia.org\/wiki\/Reed%E2%80%93Muller_code \"Reed–Muller code\").[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-deep-space-codes-18) The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a [bell curve](https:\/\/en.wikipedia.org\/wiki\/Gaussian_function \"Gaussian function\")), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977.\n\nThe [Voyager 1](https:\/\/en.wikipedia.org\/wiki\/Voyager_1 \"Voyager 1\") and [Voyager 2](https:\/\/en.wikipedia.org\/wiki\/Voyager_2 \"Voyager 2\") missions, which started in 1977, were designed to deliver color imaging and scientific information from [Jupiter](https:\/\/en.wikipedia.org\/wiki\/Jupiter \"Jupiter\") and [Saturn](https:\/\/en.wikipedia.org\/wiki\/Saturn \"Saturn\").[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-voyager-19) This resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally [Viterbi-decoded](https:\/\/en.wikipedia.org\/wiki\/Viterbi_decoder \"Viterbi decoder\")) convolutional codes that could be [concatenated](https:\/\/en.wikipedia.org\/wiki\/Concatenated_code \"Concatenated code\") with an outer [Golay (24,12,8) code](https:\/\/en.wikipedia.org\/wiki\/Binary_Golay_code \"Binary Golay code\"). The Voyager 2 craft additionally supported an implementation of a [Reed–Solomon code](https:\/\/en.wikipedia.org\/wiki\/Reed%E2%80%93Solomon_code \"Reed–Solomon code\"). The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft's extended journey to [Uranus](https:\/\/en.wikipedia.org\/wiki\/Uranus \"Uranus\") and [Neptune](https:\/\/en.wikipedia.org\/wiki\/Neptune \"Neptune\"). After ECC system upgrades in 1989, both crafts used V2 RSV coding.\n\nThe [Consultative Committee for Space Data Systems](https:\/\/en.wikipedia.org\/wiki\/Consultative_Committee_for_Space_Data_Systems \"Consultative Committee for Space Data Systems\") currently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as [Turbo codes](https:\/\/en.wikipedia.org\/wiki\/Turbo_code \"Turbo code\") or [LDPC codes](https:\/\/en.wikipedia.org\/wiki\/LDPC_code \"LDPC code\").\n\nThe different kinds of deep space and orbital missions that are conducted suggest that trying to find a one-size-fits-all error correction system will be an ongoing problem. For missions close to Earth, the nature of the [noise](https:\/\/en.wikipedia.org\/wiki\/Noise_\\(electronics\\) \"Noise (electronics)\") in the [communication channel](https:\/\/en.wikipedia.org\/wiki\/Communication_channel \"Communication channel\") is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult.\n\n### Satellite broadcasting\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=20 \"Edit section: Satellite broadcasting\")\\]\n\nThe demand for satellite [transponder](https:\/\/en.wikipedia.org\/wiki\/Transponder \"Transponder\") bandwidth continues to grow, fueled by the desire to deliver television (including new channels and [high-definition television](https:\/\/en.wikipedia.org\/wiki\/High-definition_television \"High-definition television\")) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected [modulation](https:\/\/en.wikipedia.org\/wiki\/Modulation \"Modulation\") scheme and the proportion of capacity consumed by FEC.\n\nError detection and correction codes are often used to improve the reliability of data storage media.[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-20) A parity track capable of detecting single-bit errors was present on the first [magnetic tape data storage](https:\/\/en.wikipedia.org\/wiki\/Magnetic_tape_data_storage \"Magnetic tape data storage\") in 1951. The [optimal rectangular code](https:\/\/en.wikipedia.org\/wiki\/Optimal_rectangular_code \"Optimal rectangular code\") used in [group coded recording](https:\/\/en.wikipedia.org\/wiki\/Group_coded_recording \"Group coded recording\") tapes not only detects but also corrects single-bit errors. Some [file formats](https:\/\/en.wikipedia.org\/wiki\/File_format \"File format\"), particularly [archive formats](https:\/\/en.wikipedia.org\/wiki\/Archive_formats \"Archive formats\"), include a checksum (most often CRC-32\\]) to detect corruption and truncation and can employ redundancy or [parity files](https:\/\/en.wikipedia.org\/wiki\/Parity_file \"Parity file\") to recover portions of corrupted data. [Reed-Solomon codes](https:\/\/en.wikipedia.org\/wiki\/Reed-Solomon_code \"Reed-Solomon code\") are used in [compact discs](https:\/\/en.wikipedia.org\/wiki\/Compact_disc \"Compact disc\") to correct errors caused by scratches.\n\nModern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors.[\\[21\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-21) [RAID](https:\/\/en.wikipedia.org\/wiki\/RAID \"RAID\") systems use a variety of error correction techniques to recover data when a hard drive completely fails. Filesystems such as [ZFS](https:\/\/en.wikipedia.org\/wiki\/ZFS \"ZFS\") or [Btrfs](https:\/\/en.wikipedia.org\/wiki\/Btrfs \"Btrfs\"), as well as some [RAID](https:\/\/en.wikipedia.org\/wiki\/RAID \"RAID\") implementations, support [data scrubbing](https:\/\/en.wikipedia.org\/wiki\/Data_scrubbing \"Data scrubbing\") and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used.[\\[22\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-22) The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware.\n\n### Error-correcting memory\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Error_detection_and_correction&action=edit&section=22 \"Edit section: Error-correcting memory\")\\]\n\n[Dynamic random-access memory](https:\/\/en.wikipedia.org\/wiki\/Dynamic_random-access_memory \"Dynamic random-access memory\") (DRAM) may provide stronger protection against [soft errors](https:\/\/en.wikipedia.org\/wiki\/Soft_error \"Soft error\") by relying on error-correcting codes. Such error-correcting memory, known as *ECC* or *EDAC-protected* memory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased [radiation](https:\/\/en.wikipedia.org\/wiki\/Cosmic_ray \"Cosmic ray\") in space.\n\nError-correcting memory controllers traditionally use [Hamming codes](https:\/\/en.wikipedia.org\/wiki\/Hamming_code \"Hamming code\"), although some use [triple modular redundancy](https:\/\/en.wikipedia.org\/wiki\/Triple_modular_redundancy \"Triple modular redundancy\"). [Interleaving](https:\/\/en.wikipedia.org\/wiki\/Error_correction_code#Interleaving \"Error correction code\") allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a [single-event upset](https:\/\/en.wikipedia.org\/wiki\/Single-event_upset \"Single-event upset\") (SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained.[\\[23\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-23)\n\nIn addition to hardware providing features required for ECC memory to operate, [operating systems](https:\/\/en.wikipedia.org\/wiki\/Operating_system \"Operating system\") usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the [Linux kernel](https:\/\/en.wikipedia.org\/wiki\/Linux_kernel \"Linux kernel\")'s *EDAC* subsystem (previously known as *Bluesmoke*), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the [PCI bus](https:\/\/en.wikipedia.org\/wiki\/PCI_bus \"PCI bus\").[\\[24\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-24)[\\[25\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-25)[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_note-26) A few systems\\[*[specify](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citing_sources \"Wikipedia:Citing sources\")*\\] also support [memory scrubbing](https:\/\/en.wikipedia.org\/wiki\/Memory_scrubbing \"Memory scrubbing\") to catch and correct errors early before they become unrecoverable.\n\n- [Berger code](https:\/\/en.wikipedia.org\/wiki\/Berger_code \"Berger code\")\n- [Burst error-correcting code](https:\/\/en.wikipedia.org\/wiki\/Burst_error-correcting_code \"Burst error-correcting code\")\n- [ECC memory](https:\/\/en.wikipedia.org\/wiki\/ECC_memory \"ECC memory\"), a type of computer data storage\n- [Link adaptation](https:\/\/en.wikipedia.org\/wiki\/Link_adaptation \"Link adaptation\")\n- [List of algorithms § Error detection and correction](https:\/\/en.wikipedia.org\/wiki\/List_of_algorithms#Error_detection_and_correction \"List of algorithms\")\n- [List of hash functions](https:\/\/en.wikipedia.org\/wiki\/List_of_hash_functions \"List of hash functions\")\n\n1. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-Jewish_1-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-Jewish_1-1)\n   \"Masorah\". [*Jewish Encyclopedia*](http:\/\/www.jewishencyclopedia.com\/articles\/10465-masorah).\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-2)**\n   Pratico, Gary D.; Pelt, Miles V. Van (2009). [*Basics of Biblical Hebrew Grammar: Second Edition*](https:\/\/books.google.com\/books?id=p3FIoT0s3yYC&pg=PT448). Zondervan. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-310-55882-8](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-310-55882-8 \"Special:BookSources\/978-0-310-55882-8\")\n   \n   .\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-3)**\n   Mounce, William D. (2007). [*Greek for the Rest of Us: Using Greek Tools Without Mastering Biblical Languages*](https:\/\/books.google.com\/books?id=AF-5ptJ0l2gC&pg=PA289). Zondervan. p. 289. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [978-0-310-28289-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-310-28289-1 \"Special:BookSources\/978-0-310-28289-1\")\n   \n   .\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-4)**\n   [Mishneh Torah](https:\/\/en.wikipedia.org\/wiki\/Mishneh_Torah \"Mishneh Torah\"), Tefillin, Mezuzah, and Sefer Torah, 1:2. Example English translation:\n   \n   [Eliyahu Touger](https:\/\/en.wikipedia.org\/w\/index.php?title=Eliyahu_Touger&action=edit&redlink=1 \"Eliyahu Touger (page does not exist)\"). [*The Rambam's Mishneh Torah*](https:\/\/www.chabad.org\/library\/article_cdo\/aid\/925417\/jewish\/Tefillin-Mezuzah-and-Sefer-Torah-Chapter-One.htm). [Moznaim Publishing Corporation](https:\/\/en.wikipedia.org\/w\/index.php?title=Moznaim_Publishing_Corporation&action=edit&redlink=1 \"Moznaim Publishing Corporation (page does not exist)\").\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-5)**\n   Brian M. Fagan (5 December 1996). \"Dead Sea Scrolls\". *[The Oxford Companion to Archaeology](https:\/\/en.wikipedia.org\/w\/index.php?title=The_Oxford_Companion_to_Archaeology&action=edit&redlink=1 \"The Oxford Companion to Archaeology (page does not exist)\")*. [Oxford University Press](https:\/\/en.wikipedia.org\/wiki\/Oxford_University_Press \"Oxford University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0195076184](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0195076184 \"Special:BookSources\/0195076184\")\n   \n   .\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-Thompson_6-0)**\n   Thompson, Thomas M. (1983), *From Error-Correcting Codes through Sphere Packings to Simple Groups*, The Carus Mathematical Monographs (\\#21), The Mathematical Association of America, p. vii, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-88385-023-0](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-88385-023-0 \"Special:BookSources\/0-88385-023-0\")\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-7)**\n   Shannon, C.E. (1948), \"A Mathematical Theory of Communication\", *Bell System Technical Journal*, **27** (3): 379–423, [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1002\/j.1538-7305.1948.tb01338.x](https:\/\/doi.org\/10.1002%2Fj.1538-7305.1948.tb01338.x), [hdl](https:\/\/en.wikipedia.org\/wiki\/Hdl_\\(identifier\\) \"Hdl (identifier)\"):[10338\\.dmlcz\/101429](https:\/\/hdl.handle.net\/10338.dmlcz%2F101429), [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [9230594](https:\/\/pubmed.ncbi.nlm.nih.gov\/9230594)\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-8)**\n   Golay, Marcel J. E. (1949), \"Notes on Digital Coding\", *Proc.I.R.E. (I.E.E.E.)*, **37**: 657\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-9)**\n   Gupta, Vikas; Verma, Chanderkant (November 2012). \"Error Detection and Correction: An Introduction\". *International Journal of Advanced Research in Computer Science and Software Engineering*. **2** (11). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [17499858](https:\/\/api.semanticscholar.org\/CorpusID:17499858).\n10. ^ [***a***](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-reliable-erasure-code_10-0) [***b***](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-reliable-erasure-code_10-1) A. J. McAuley, *Reliable Broadband Communication Using a Burst Erasure Correcting Code*, ACM SIGCOMM, 1990.\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-11)**\n    Shah, Pradeep M.; Vyavahare, Prakash D.; Jain, Anjana (September 2015). [\"Modern error correcting codes for 4G and beyond: Turbo codes and LDPC codes\"](https:\/\/www.researchgate.net\/publication\/301611980). *2015 Radio and Antenna Days of the Indian Ocean (RADIO)*. pp. 1–2\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1109\/RADIO.2015.7323369](https:\/\/doi.org\/10.1109%2FRADIO.2015.7323369). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-9-9903-7339-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-9-9903-7339-4 \"Special:BookSources\/978-9-9903-7339-4\")\n    \n    . [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [28885076](https:\/\/api.semanticscholar.org\/CorpusID:28885076). Retrieved 22 May 2022.\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-12)**\n    [\"IEEE SA - IEEE 802.11ac-2013\"](https:\/\/web.archive.org\/web\/20220522205452\/https:\/\/standards.ieee.org\/ieee\/802.11ac\/4473\/). *IEEE Standards Association*. Archived from [the original](https:\/\/standards.ieee.org\/ieee\/802.11ac\/4473\/) on 2022-05-22. Retrieved 2022-05-22.\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-13)**\n    [\"Transition to Advanced Format 4K Sector Hard Drives \\| Seagate US\"](https:\/\/www.seagate.com\/sg\/en\/tech-insights\/advanced-format-4k-sector-hard-drives-master-ti\/). *Seagate.com*. Retrieved 22 May 2022.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-14)**\n    Frank van Gerwen. [\"Numbers (and other mysterious) stations\"](https:\/\/web.archive.org\/web\/20170712053335\/http:\/\/www.cisquet.nl\/numbers.htm). Archived from [the original](http:\/\/www.cisquet.nl\/numbers.htm) on 12 July 2017. Retrieved 12 March 2012.\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-15)**\n    Gary Cutlack (25 August 2010). [\"Mysterious Russian 'Numbers Station' Changes Broadcast After 20 Years\"](https:\/\/web.archive.org\/web\/20170705131816\/https:\/\/www.gizmodo.com.au\/2010\/08\/mysterious-russian-numbers-station-changes-broadcast-after-20-years\/). *Gizmodo*. Archived from [the original](http:\/\/www.gizmodo.com.au\/2010\/08\/mysterious-russian-numbers-station-changes-broadcast-after-20-years\/) on 5 July 2017. Retrieved 12 March 2012.\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-16)**\n    Ben-Gal I.; Herer Y.; Raz T. (2003). [\"Self-correcting inspection procedure under inspection errors\"](https:\/\/web.archive.org\/web\/20131013171945\/http:\/\/www.eng.tau.ac.il\/~bengal\/SCI_paper.pdf) (PDF). *IIE Transactions*. IIE Transactions on Quality and Reliability, 34(6), pp. 529-540. Archived from [the original](http:\/\/www.eng.tau.ac.il\/~bengal\/SCI_paper.pdf) (PDF) on 2013-10-13. Retrieved 2014-01-10.\n17. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-DOI10.7275\/bjvb-2n37_17-0)**\n    Yvo Meeres, Tommi A. Pirinen (2021). \"Vowel Harmony Viewed as Error-Correcting Code\". *Proceedings of the Society for Computation in Linguistics*. **4** (1): 313–322\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.7275\/bjvb-2n37](https:\/\/doi.org\/10.7275%2Fbjvb-2n37).\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-deep-space-codes_18-0)** K. Andrews et al., *The Development of Turbo and LDPC Codes for Deep-Space Applications*, Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007.\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-voyager_19-0)**\n    Huffman, William Cary; [Pless, Vera S.](https:\/\/en.wikipedia.org\/wiki\/Vera_Pless \"Vera Pless\") (2003). [*Fundamentals of Error-Correcting Codes*](https:\/\/archive.org\/details\/fundamentalsofer0000huff). [Cambridge University Press](https:\/\/en.wikipedia.org\/wiki\/Cambridge_University_Press \"Cambridge University Press\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-521-78280-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-521-78280-7 \"Special:BookSources\/978-0-521-78280-7\")\n    \n    .\n20. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-20)**\n    Kurtas, Erozan M.; Vasic, Bane (2018-10-03). *Advanced Error Control Techniques for Data Storage Systems*. CRC Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4200-3649-7](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4200-3649-7 \"Special:BookSources\/978-1-4200-3649-7\")\n    \n    .\n21. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-21)**\n    Scott A. Moulton. [\"My Hard Drive Died\"](https:\/\/web.archive.org\/web\/20080202143103\/http:\/\/www.myharddrivedied.com\/presentations_whitepaper.html). Archived from [the original](http:\/\/www.myharddrivedied.com\/presentations_whitepaper.html) on 2008-02-02.\n22. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-22)**\n    Qiao, Zhi; Fu, Song; Chen, Hsing-Bung; Settlemyer, Bradley (2019). \"Building Reliable High-Performance Storage Systems: An Empirical and Analytical Study\". *2019 IEEE International Conference on Cluster Computing (CLUSTER)*. pp. 1–10\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1109\/CLUSTER.2019.8891006](https:\/\/doi.org\/10.1109%2FCLUSTER.2019.8891006). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-7281-4734-5](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-7281-4734-5 \"Special:BookSources\/978-1-7281-4734-5\")\n    \n    . [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [207951690](https:\/\/api.semanticscholar.org\/CorpusID:207951690).\n23. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-23)**\n    [\"Using StrongArm SA-1110 in the On-Board Computer of Nanosatellite\"](https:\/\/web.archive.org\/web\/20111002152735\/http:\/\/www.apmcsta.org\/File\/doc\/Conferences\/6th%20meeting\/Chen%20Zhenyu.doc). Tsinghua Space Center, [Tsinghua University](https:\/\/en.wikipedia.org\/wiki\/Tsinghua_University \"Tsinghua University\"), Beijing. Archived from [the original](http:\/\/www.apmcsta.org\/File\/doc\/Conferences\/6th%20meeting\/Chen%20Zhenyu.doc) on 2011-10-02. Retrieved 2009-02-16.\n24. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-24)**\n    Jeff Layton. [\"Error Detection and Correction\"](http:\/\/www.admin-magazine.com\/Articles\/Monitoring-Memory-Errors). *[Linux Magazine](https:\/\/en.wikipedia.org\/wiki\/Linux_Magazine \"Linux Magazine\")*. Retrieved 2014-08-12.\n25. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-25)**\n    [\"EDAC Project\"](https:\/\/bluesmoke.sourceforge.net\/). *bluesmoke.sourceforge.net*. Retrieved 2014-08-12.\n26. **[^](https:\/\/en.wikipedia.org\/wiki\/Error_detection_and_correction#cite_ref-26)**\n    [\"Documentation\/edac.txt\"](https:\/\/web.archive.org\/web\/20090905174616\/http:\/\/www.kernel.org\/doc\/Documentation\/edac.txt). *Linux kernel documentation*. [kernel.org](https:\/\/en.wikipedia.org\/wiki\/Kernel.org \"Kernel.org\"). 2014-06-16. Archived from [the original](https:\/\/www.kernel.org\/doc\/Documentation\/edac.txt) on 2009-09-05. Retrieved 2014-08-12.\n\n- Shu Lin; Daniel J. Costello, Jr. (1983). *Error Control Coding: Fundamentals and Applications*. [Prentice Hall](https:\/\/en.wikipedia.org\/wiki\/Prentice_Hall \"Prentice Hall\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  \n  [0-13-283796-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-13-283796-X \"Special:BookSources\/0-13-283796-X\")\n  \n  .\n- [SoftECC: A System for Software Memory Integrity Checking](http:\/\/pdos.csail.mit.edu\/papers\/softecc:ddopson-meng\/softecc_ddopson-meng.pdf)\n- [A Tunable, Software-based DRAM Error Detection and Correction Library for HPC](http:\/\/www.fiala.me\/pubs\/papers\/libsdc11.pdf) [Archived](https:\/\/web.archive.org\/web\/20141107074502\/http:\/\/www.fiala.me\/pubs\/papers\/libsdc11.pdf) 2014-11-07 at the [Wayback Machine](https:\/\/en.wikipedia.org\/wiki\/Wayback_Machine \"Wayback Machine\")\n- [Detection and Correction of Silent Data Corruption for Large-Scale High-Performance Computing](http:\/\/www.fiala.me\/pubs\/papers\/sc12-redmpi.pdf) [Archived](https:\/\/web.archive.org\/web\/20141107074511\/http:\/\/www.fiala.me\/pubs\/papers\/sc12-redmpi.pdf) 2014-11-07 at the [Wayback Machine](https:\/\/en.wikipedia.org\/wiki\/Wayback_Machine \"Wayback Machine\")\n\n- [The on-line textbook: Information Theory, Inference, and Learning Algorithms](http:\/\/www.inference.phy.cam.ac.uk\/mackay\/itila\/), by [David J.C. MacKay](https:\/\/en.wikipedia.org\/wiki\/David_J.C._MacKay \"David J.C. MacKay\"), contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including [low-density parity-check codes](https:\/\/en.wikipedia.org\/wiki\/Low-density_parity-check_code \"Low-density parity-check code\"), [turbo codes](https:\/\/en.wikipedia.org\/wiki\/Turbo_code \"Turbo code\"), and [fountain codes](https:\/\/en.wikipedia.org\/wiki\/Fountain_codes \"Fountain codes\").\n- [ECC Page](http:\/\/www.eccpage.com\/) – implementations of popular ECC encoding and decoding routines","meta_canonical":null}

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Boilerpipe Text	To clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied Reed–Solomon error correction (right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted. In information theory and coding theory with applications in computer science and telecommunications , error detection and correction ( EDAC ) or error control are techniques that enable reliable delivery of digital data over unreliable communication channels . Many communication channels are subject to channel noise , and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. Error detection is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. Error correction is the detection of errors and reconstruction of the original, error-free data. In classical antiquity, copyists of the Hebrew Bible were paid for their work according to the number of stichs (lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters. [ 1 ] This also helped ensure accuracy in the transmission of the text with the production of subsequent copies. [ 2 ] [ 3 ] Between the 7th and 10th centuries CE a group of Jewish scribes formalized and expanded this to create the Numerical Masorah to ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary. [ 1 ] Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable. [ 4 ] The effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the Dead Sea Scrolls in 1947–1956, dating from c. 150 BCE – 75 CE . [ 5 ] The modern development of error correction codes is credited to Richard Hamming in 1947. [ 6 ] A description of Hamming's code appeared in Claude Shannon 's A Mathematical Theory of Communication [ 7 ] and was quickly generalized by Marcel J. E. Golay . [ 8 ] All error-detection and correction schemes add some redundancy (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted. Error detection and correction schemes can be either systematic or non-systematic. In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of check bits (or parity data ), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message. Good error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common channel models include memoryless models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in bursts . Consequently, error-detecting and -correcting codes can be generally distinguished between random-error-detecting/correcting and burst-error-detecting/correcting . Some codes can also be suitable for a mixture of random errors and burst errors. If the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as automatic repeat request (ARQ), and is most notably used in the Internet. An alternate approach for error control is hybrid automatic repeat request (HARQ), which is a combination of ARQ and error-correction coding. Types of error correction [ edit ] There are three major types of error correction: [ 9 ] Automatic repeat request [ edit ] Automatic repeat request (ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and/or negative acknowledgment messages, and timeouts to achieve reliable data transmission. An acknowledgment is a message sent by the receiver to indicate that it has correctly received a data frame . Usually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e., within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions. Three types of ARQ protocols are Stop-and-wait ARQ , Go-Back-N ARQ , and Selective Repeat ARQ . ARQ is appropriate if the communication channel has varying or unknown capacity , such as is the case on the Internet. However, ARQ requires the availability of a back channel , results in possibly increased latency due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of network congestion can put a strain on the server and overall network capacity. [ 10 ] For example, ARQ is used on shortwave radio data links in the form of ARQ-E , or combined with multiplexing as ARQ-M . Forward error correction [ edit ] Forward error correction (FEC) is a process of adding redundant data such as an error-correcting code (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a backchannel is not required in forward error correction. Error-correcting codes are used in lower-layer communication such as cellular network , high-speed fiber-optic communication and Wi-Fi , [ 11 ] [ 12 ] as well as for reliable storage in media such as flash memory , hard disk and RAM . [ 13 ] Error-correcting codes are usually distinguished between convolutional codes and block codes : Convolutional codes are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the Viterbi decoder allows optimal decoding . Block codes are processed on a block-by-block basis. Early examples of block codes are repetition codes , Hamming codes and multidimensional parity-check codes . They were followed by a number of efficient codes, Reed–Solomon codes being the most notable due to their current widespread use. Turbo codes and low-density parity-check codes (LDPC) are relatively new constructions that can provide almost optimal efficiency . Shannon's theorem is an important theorem in forward error correction, and describes the maximum information rate at which reliable communication is possible over a channel that has a certain error probability or signal-to-noise ratio (SNR). This strict upper limit is expressed in terms of the channel capacity . More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a discrete memoryless channel can be made arbitrarily small, provided that the code rate is smaller than the channel capacity. The code rate is defined as the fraction k/n of k source symbols and n encoded symbols. The actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have efficient encoding and decoding algorithms. Hybrid ARQ is a combination of ARQ and forward error correction. There are two basic approaches: [ 10 ] Messages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check). Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message. The latter approach is particularly attractive on an erasure channel when using a rateless erasure code . Types of error detection [ edit ] Error detection is most commonly realized using a suitable hash function (or specifically, a checksum , cyclic redundancy check or other algorithm). A hash function adds a fixed-length tag to a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided. There exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check's performance in detecting burst errors ). Minimum distance coding [ edit ] A random-error-correcting code based on minimum distance coding can provide a strict guarantee on the number of detectable errors, but it may not protect against a preimage attack . A repetition code is a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern 1011 , the four-bit block can be repeated three times, thus producing 1011 1011 1011 . If this twelve-bit pattern was received as 1010 1011 1011 – where the first block is unlike the other two – an error has occurred. A repetition code is very inefficient and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g., 1010 1010 1010 in the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of numbers stations . [ 14 ] [ 15 ] A parity bit is a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous. Parity bits added to each word sent are called transverse redundancy checks , while those added at the end of a stream of words are called longitudinal redundancy checks . For example, if each of a series of m-bit words has a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words. With more error correction bits, more errors can be detected and in some cases corrected. There are also other bit-grouping techniques. A checksum of a message is a modular arithmetic sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a ones'-complement operation prior to transmission to detect unintentional all-zero messages. Checksum schemes include parity bits, check digits , and longitudinal redundancy checks . Some checksum schemes, such as the Damm algorithm , the Luhn algorithm , and the Verhoeff algorithm , are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers. Cyclic redundancy check [ edit ] A cyclic redundancy check (CRC) is a non-secure hash function designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a generator polynomial , which is used as the divisor in a polynomial long division over a finite field , taking the input data as the dividend . The remainder becomes the result. A CRC has properties that make it well suited for detecting burst errors . CRCs are particularly easy to implement in hardware and are therefore commonly used in computer networks and storage devices such as hard disk drives . The parity bit can be seen as a special-case 1-bit CRC. Cryptographic hash function [ edit ] The output of a cryptographic hash function , also known as a message digest , can provide strong assurances about data integrity , whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a keyed hash or message authentication code (MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message. Digital signatures can provide strong assurances about data integrity, whether the changes of the data are accidental or maliciously introduced. Digital signatures are perhaps most notable for being part of the HTTPS protocol for securely browsing the web. Error correction code [ edit ] Any error-correcting code can be used for error detection. A code with minimum Hamming distance , d , can detect up to d − 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired. Codes with minimum Hamming distance d = 2 are degenerate cases of error-correcting codes and can be used to detect single errors. The parity bit is an example of a single-error-detecting code. Applications that require low latency (such as telephone conversations) cannot use automatic repeat request (ARQ); they must use forward error correction (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be usable. Applications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available. Applications that use ARQ must have a return channel ; applications having no return channel cannot use ARQ. Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC. Reliability and inspection engineering also make use of the theory of error-correcting codes, [ 16 ] as well as natural language. [ 17 ] In a typical TCP/IP stack, error control is performed at multiple levels: Each Ethernet frame uses CRC-32 error detection. Frames with detected errors are discarded by the receiver hardware. The IPv4 header contains a checksum protecting the contents of the header. Packets with incorrect checksums are dropped within the network or at the receiver. The checksum was omitted from the IPv6 header in order to minimize processing costs in network routing and because current link layer technology is assumed to provide sufficient error detection (see also RFC 3819). UDP has an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the network stack . The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection. TCP provides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack and eventually get retransmitted using ARQ, either explicitly (such as through three-way handshake ) or implicitly due to a timeout . Deep-space telecommunications [ edit ] The development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded) convolutional codes and Reed–Muller codes . [ 18 ] The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a bell curve ), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977. The Voyager 1 and Voyager 2 missions, which started in 1977, were designed to deliver color imaging and scientific information from Jupiter and Saturn . [ 19 ] This resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally Viterbi-decoded ) convolutional codes that could be concatenated with an outer Golay (24,12,8) code . The Voyager 2 craft additionally supported an implementation of a Reed–Solomon code . The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft's extended journey to Uranus and Neptune . After ECC system upgrades in 1989, both crafts used V2 RSV coding. The Consultative Committee for Space Data Systems currently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as Turbo codes or LDPC codes . The different kinds of deep space and orbital missions that are conducted suggest that trying to find a one-size-fits-all error correction system will be an ongoing problem. For missions close to Earth, the nature of the noise in the communication channel is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult. Satellite broadcasting [ edit ] The demand for satellite transponder bandwidth continues to grow, fueled by the desire to deliver television (including new channels and high-definition television ) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected modulation scheme and the proportion of capacity consumed by FEC. Error detection and correction codes are often used to improve the reliability of data storage media. [ 20 ] A parity track capable of detecting single-bit errors was present on the first magnetic tape data storage in 1951. The optimal rectangular code used in group coded recording tapes not only detects but also corrects single-bit errors. Some file formats , particularly archive formats , include a checksum (most often CRC-32]) to detect corruption and truncation and can employ redundancy or parity files to recover portions of corrupted data. Reed-Solomon codes are used in compact discs to correct errors caused by scratches. Modern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors. [ 21 ] RAID systems use a variety of error correction techniques to recover data when a hard drive completely fails. Filesystems such as ZFS or Btrfs , as well as some RAID implementations, support data scrubbing and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used. [ 22 ] The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware. Error-correcting memory [ edit ] Dynamic random-access memory (DRAM) may provide stronger protection against soft errors by relying on error-correcting codes. Such error-correcting memory, known as ECC or EDAC-protected memory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased radiation in space. Error-correcting memory controllers traditionally use Hamming codes , although some use triple modular redundancy . Interleaving allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a single-event upset (SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained. [ 23 ] In addition to hardware providing features required for ECC memory to operate, operating systems usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the Linux kernel 's EDAC subsystem (previously known as Bluesmoke ), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the PCI bus . [ 24 ] [ 25 ] [ 26 ] A few systems [ specify ] also support memory scrubbing to catch and correct errors early before they become unrecoverable. Berger code Burst error-correcting code ECC memory , a type of computer data storage Link adaptation List of algorithms § Error detection and correction List of hash functions ^ a b "Masorah". Jewish Encyclopedia . ^ Pratico, Gary D.; Pelt, Miles V. Van (2009). Basics of Biblical Hebrew Grammar: Second Edition . Zondervan. ISBN 978-0-310-55882-8 . ^ Mounce, William D. (2007). Greek for the Rest of Us: Using Greek Tools Without Mastering Biblical Languages . Zondervan. p. 289. ISBN 978-0-310-28289-1 . ^ Mishneh Torah , Tefillin, Mezuzah, and Sefer Torah, 1:2. Example English translation: Eliyahu Touger . The Rambam's Mishneh Torah . Moznaim Publishing Corporation . ^ Brian M. Fagan (5 December 1996). "Dead Sea Scrolls". The Oxford Companion to Archaeology . Oxford University Press . ISBN 0195076184 . ^ Thompson, Thomas M. (1983), From Error-Correcting Codes through Sphere Packings to Simple Groups , The Carus Mathematical Monographs (#21), The Mathematical Association of America, p. vii, ISBN 0-88385-023-0 ^ Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal , 27 (3): 379– 423, doi : 10.1002/j.1538-7305.1948.tb01338.x , hdl : 10338.dmlcz/101429 , PMID 9230594 ^ Golay, Marcel J. E. (1949), "Notes on Digital Coding", Proc.I.R.E. (I.E.E.E.) , 37 : 657 ^ Gupta, Vikas; Verma, Chanderkant (November 2012). "Error Detection and Correction: An Introduction". International Journal of Advanced Research in Computer Science and Software Engineering . 2 (11). S2CID 17499858 . ^ a b A. J. McAuley, Reliable Broadband Communication Using a Burst Erasure Correcting Code , ACM SIGCOMM, 1990. ^ Shah, Pradeep M.; Vyavahare, Prakash D.; Jain, Anjana (September 2015). "Modern error correcting codes for 4G and beyond: Turbo codes and LDPC codes" . 2015 Radio and Antenna Days of the Indian Ocean (RADIO) . pp. 1– 2. doi : 10.1109/RADIO.2015.7323369 . ISBN 978-9-9903-7339-4 . S2CID 28885076 . Retrieved 22 May 2022 . ^ "IEEE SA - IEEE 802.11ac-2013" . IEEE Standards Association . Archived from the original on 2022-05-22 . Retrieved 2022-05-22 . ^ "Transition to Advanced Format 4K Sector Hard Drives \| Seagate US" . Seagate.com . Retrieved 22 May 2022 . ^ Frank van Gerwen. "Numbers (and other mysterious) stations" . Archived from the original on 12 July 2017 . Retrieved 12 March 2012 . ^ Gary Cutlack (25 August 2010). "Mysterious Russian 'Numbers Station' Changes Broadcast After 20 Years" . Gizmodo . Archived from the original on 5 July 2017 . Retrieved 12 March 2012 . ^ Ben-Gal I.; Herer Y.; Raz T. (2003). "Self-correcting inspection procedure under inspection errors" (PDF) . IIE Transactions . IIE Transactions on Quality and Reliability, 34(6), pp. 529-540. Archived from the original (PDF) on 2013-10-13 . Retrieved 2014-01-10 . ^ Yvo Meeres, Tommi A. Pirinen (2021). "Vowel Harmony Viewed as Error-Correcting Code". Proceedings of the Society for Computation in Linguistics . 4 (1): 313– 322. doi : 10.7275/bjvb-2n37 . ^ K. Andrews et al., The Development of Turbo and LDPC Codes for Deep-Space Applications , Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007. ^ Huffman, William Cary; Pless, Vera S. (2003). Fundamentals of Error-Correcting Codes . Cambridge University Press . ISBN 978-0-521-78280-7 . ^ Kurtas, Erozan M.; Vasic, Bane (2018-10-03). Advanced Error Control Techniques for Data Storage Systems . CRC Press. ISBN 978-1-4200-3649-7 . ^ Scott A. Moulton. "My Hard Drive Died" . Archived from the original on 2008-02-02. ^ Qiao, Zhi; Fu, Song; Chen, Hsing-Bung; Settlemyer, Bradley (2019). "Building Reliable High-Performance Storage Systems: An Empirical and Analytical Study". 2019 IEEE International Conference on Cluster Computing (CLUSTER) . pp. 1– 10. doi : 10.1109/CLUSTER.2019.8891006 . ISBN 978-1-7281-4734-5 . S2CID 207951690 . ^ "Using StrongArm SA-1110 in the On-Board Computer of Nanosatellite" . Tsinghua Space Center, Tsinghua University , Beijing. Archived from the original on 2011-10-02 . Retrieved 2009-02-16 . ^ Jeff Layton. "Error Detection and Correction" . Linux Magazine . Retrieved 2014-08-12 . ^ "EDAC Project" . bluesmoke.sourceforge.net . Retrieved 2014-08-12 . ^ "Documentation/edac.txt" . Linux kernel documentation . kernel.org . 2014-06-16. Archived from the original on 2009-09-05 . Retrieved 2014-08-12 . Shu Lin; Daniel J. Costello, Jr. (1983). Error Control Coding: Fundamentals and Applications . Prentice Hall . ISBN 0-13-283796-X . SoftECC: A System for Software Memory Integrity Checking A Tunable, Software-based DRAM Error Detection and Correction Library for HPC Archived 2014-11-07 at the Wayback Machine Detection and Correction of Silent Data Corruption for Large-Scale High-Performance Computing Archived 2014-11-07 at the Wayback Machine The on-line textbook: Information Theory, Inference, and Learning Algorithms , by David J.C. MacKay , contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including low-density parity-check codes , turbo codes , and fountain codes . ECC Page – implementations of popular ECC encoding and decoding routines
Markdown	[Jump to content](https://en.wikipedia.org/wiki/Error_detection_and_correction#bodyContent) Main menu Main menu move to sidebar hide Navigation - [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]") - [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia") - [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events") - [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]") - [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works") - [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia") Contribute - [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia") - [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia") - [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors") - [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]") - [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia") - [Special pages](https://en.wikipedia.org/wiki/Special:SpecialPages "A list of all special pages [q]") [![](https://en.wikipedia.org/static/images/icons/enwiki-25.svg) ![Wikipedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en-25.svg)](https://en.wikipedia.org/wiki/Main_Page) [Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]") Appearance - [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en) - [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Error+detection+and+correction "You are encouraged to create an account and log in; however, it is not mandatory") - [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Error+detection+and+correction "You're encouraged to log in; however, it's not mandatory. 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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Error_detection_and_correction) - [1 Definitions](https://en.wikipedia.org/wiki/Error_detection_and_correction#Definitions) - [2 History](https://en.wikipedia.org/wiki/Error_detection_and_correction#History) - [3 Principles](https://en.wikipedia.org/wiki/Error_detection_and_correction#Principles) - [4 Types of error correction](https://en.wikipedia.org/wiki/Error_detection_and_correction#Types_of_error_correction) Toggle Types of error correction subsection - [4\.1 Automatic repeat request](https://en.wikipedia.org/wiki/Error_detection_and_correction#Automatic_repeat_request) - [4\.2 Forward error correction](https://en.wikipedia.org/wiki/Error_detection_and_correction#Forward_error_correction) - [4\.3 Hybrid schemes](https://en.wikipedia.org/wiki/Error_detection_and_correction#Hybrid_schemes) - [5 Types of error detection](https://en.wikipedia.org/wiki/Error_detection_and_correction#Types_of_error_detection) Toggle Types of error detection subsection - [5\.1 Minimum distance coding](https://en.wikipedia.org/wiki/Error_detection_and_correction#Minimum_distance_coding) - [5\.2 Repetition codes](https://en.wikipedia.org/wiki/Error_detection_and_correction#Repetition_codes) - [5\.3 Parity bit](https://en.wikipedia.org/wiki/Error_detection_and_correction#Parity_bit) - [5\.4 Checksum](https://en.wikipedia.org/wiki/Error_detection_and_correction#Checksum) - [5\.5 Cyclic redundancy check](https://en.wikipedia.org/wiki/Error_detection_and_correction#Cyclic_redundancy_check) - [5\.6 Cryptographic hash function](https://en.wikipedia.org/wiki/Error_detection_and_correction#Cryptographic_hash_function) - [5\.7 Digital signature](https://en.wikipedia.org/wiki/Error_detection_and_correction#Digital_signature) - [5\.8 Error correction code](https://en.wikipedia.org/wiki/Error_detection_and_correction#Error_correction_code) - [6 Applications](https://en.wikipedia.org/wiki/Error_detection_and_correction#Applications) Toggle Applications subsection - [6\.1 Internet](https://en.wikipedia.org/wiki/Error_detection_and_correction#Internet) - [6\.2 Deep-space telecommunications](https://en.wikipedia.org/wiki/Error_detection_and_correction#Deep-space_telecommunications) - [6\.3 Satellite broadcasting](https://en.wikipedia.org/wiki/Error_detection_and_correction#Satellite_broadcasting) - [6\.4 Data storage](https://en.wikipedia.org/wiki/Error_detection_and_correction#Data_storage) - [6\.5 Error-correcting memory](https://en.wikipedia.org/wiki/Error_detection_and_correction#Error-correcting_memory) - [7 See also](https://en.wikipedia.org/wiki/Error_detection_and_correction#See_also) - [8 References](https://en.wikipedia.org/wiki/Error_detection_and_correction#References) - [9 Further reading](https://en.wikipedia.org/wiki/Error_detection_and_correction#Further_reading) - [10 External links](https://en.wikipedia.org/wiki/Error_detection_and_correction#External_links) Toggle the table of contents # Error detection and correction 36 languages - [العربية](https://ar.wikipedia.org/wiki/%D9%83%D8%B4%D9%81_%D8%A7%D9%84%D8%A3%D8%AE%D8%B7%D8%A7%D8%A1_%D9%88%D8%AA%D8%B5%D8%AD%D9%8A%D8%AD%D9%87%D8%A7 "كشف الأخطاء وتصحيحها – Arabic") - 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[Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q1062839 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Reliable digital data delivery methods on unreliable channels Not to be confused with [Exception handling](https://en.wikipedia.org/wiki/Exception_handling "Exception handling"). \| \| \| \|---\|---\| \| [![icon](https://upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/60px-Question_book-new.svg.png)](https://en.wikipedia.org/wiki/File:Question_book-new.svg) \| This article needs additional citations for [verification](https://en.wikipedia.org/wiki/Wikipedia:Verifiability "Wikipedia:Verifiability"). Please help [improve this article](https://en.wikipedia.org/wiki/Special:EditPage/Error_detection_and_correction "Special:EditPage/Error detection and correction") by [adding citations to reliable sources](https://en.wikipedia.org/wiki/Help:Referencing_for_beginners "Help:Referencing for beginners") in this article. Unsourced material may be challenged and removed. Find sources: ["Error detection and correction"](https://www.google.com/search?as_eq=wikipedia&q=%22Error+detection+and+correction%22) – [news](https://www.google.com/search?tbm=nws&q=%22Error+detection+and+correction%22+-wikipedia&tbs=ar:1) · [newspapers](https://www.google.com/search?&q=%22Error+detection+and+correction%22&tbs=bkt:s&tbm=bks) · [books](https://www.google.com/search?tbs=bks:1&q=%22Error+detection+and+correction%22+-wikipedia) · [scholar](https://scholar.google.com/scholar?q=%22Error+detection+and+correction%22) · [JSTOR](https://www.jstor.org/action/doBasicSearch?Query=%22Error+detection+and+correction%22&acc=on&wc=on) (August 2008) ([Learn how and when to remove this message](https://en.wikipedia.org/wiki/Help:Maintenance_template_removal "Help:Maintenance template removal")) \| [![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg/250px-Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg)](https://en.wikipedia.org/wiki/File:Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg) To clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied [Reed–Solomon error correction](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction "Reed–Solomon error correction") (right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted. In [information theory](https://en.wikipedia.org/wiki/Information_theory "Information theory") and [coding theory](https://en.wikipedia.org/wiki/Coding_theory "Coding theory") with applications in [computer science](https://en.wikipedia.org/wiki/Computer_science "Computer science") and [telecommunications](https://en.wikipedia.org/wiki/Telecommunications "Telecommunications"), error detection and correction (EDAC) or error control are techniques that enable [reliable delivery](https://en.wikipedia.org/wiki/Reliable_delivery "Reliable delivery") of [digital data](https://en.wikipedia.org/wiki/Digital_data "Digital data") over unreliable [communication channels](https://en.wikipedia.org/wiki/Communication_channel "Communication channel"). Many communication channels are subject to [channel noise](https://en.wikipedia.org/wiki/Channel_noise "Channel noise"), and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. ## Definitions \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=1 "Edit section: Definitions")\] Error detection is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. Error correction is the detection of errors and reconstruction of the original, error-free data. ## History \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=2 "Edit section: History")\] In classical antiquity, [copyists](https://en.wikipedia.org/wiki/Copyist "Copyist") of the [Hebrew Bible](https://en.wikipedia.org/wiki/Hebrew_Bible "Hebrew Bible") were paid for their work according to the number of [stichs](https://en.wiktionary.org/wiki/stich "wiktionary:stich") (lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters.[\[1\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-Jewish-1) This also helped ensure accuracy in the transmission of the text with the production of subsequent copies.[\[2\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-2)[\[3\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-3) Between the 7th and 10th centuries CE a [group of Jewish scribes](https://en.wikipedia.org/wiki/Masoretes "Masoretes") formalized and expanded this to create the [Numerical Masorah](https://en.wikipedia.org/wiki/Masoretic_Text#Numerical_Masorah "Masoretic Text") to ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary.[\[1\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-Jewish-1) Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable.[\[4\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-4) The effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the [Dead Sea Scrolls](https://en.wikipedia.org/wiki/Dead_Sea_Scrolls "Dead Sea Scrolls") in 1947–1956, dating from c. 150 BCE – 75 CE.[\[5\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-5) The modern development of [error correction codes](https://en.wikipedia.org/wiki/Error_correction_code "Error correction code") is credited to [Richard Hamming](https://en.wikipedia.org/wiki/Richard_Hamming "Richard Hamming") in 1947.[\[6\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-Thompson-6) A description of [Hamming's code](https://en.wikipedia.org/wiki/Hamming%27s_code "Hamming's code") appeared in [Claude Shannon](https://en.wikipedia.org/wiki/Claude_Shannon "Claude Shannon")'s A Mathematical Theory of Communication[\[7\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-7) and was quickly generalized by [Marcel J. E. Golay](https://en.wikipedia.org/wiki/Marcel_J._E._Golay "Marcel J. E. Golay").[\[8\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-8) ## Principles \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=3 "Edit section: Principles")\] All error-detection and correction schemes add some [redundancy](https://en.wikipedia.org/wiki/Redundancy_\(information_theory\) "Redundancy (information theory)") (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted. Error detection and correction schemes can be either [systematic](https://en.wikipedia.org/wiki/Systematic_code "Systematic code") or non-systematic. In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of check bits (or parity data), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message. Good error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common [channel models](https://en.wikipedia.org/wiki/Channel_model "Channel model") include [memoryless](https://en.wikipedia.org/wiki/Memoryless "Memoryless") models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in [bursts](https://en.wikipedia.org/wiki/Burst_error "Burst error"). Consequently, error-detecting and -correcting codes can be generally distinguished between random-error-detecting/correcting and burst-error-detecting/correcting. Some codes can also be suitable for a mixture of random errors and burst errors. If the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as [automatic repeat request](https://en.wikipedia.org/wiki/Automatic_repeat_request "Automatic repeat request") (ARQ), and is most notably used in the Internet. An alternate approach for error control is [hybrid automatic repeat request](https://en.wikipedia.org/wiki/Hybrid_automatic_repeat_request "Hybrid automatic repeat request") (HARQ), which is a combination of ARQ and error-correction coding. ## Types of error correction \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=4 "Edit section: Types of error correction")\] There are three major types of error correction:[\[9\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-9) ### Automatic repeat request \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=5 "Edit section: Automatic repeat request")\] [Automatic repeat request](https://en.wikipedia.org/wiki/Automatic_repeat_request "Automatic repeat request") (ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and/or negative acknowledgment messages, and [timeouts](https://en.wikipedia.org/wiki/Timeout_\(computing\) "Timeout (computing)") to achieve reliable data transmission. An acknowledgment is a message sent by the receiver to indicate that it has correctly received a [data frame](https://en.wikipedia.org/wiki/Data_frame "Data frame"). Usually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e., within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions. Three types of ARQ protocols are [Stop-and-wait ARQ](https://en.wikipedia.org/wiki/Stop-and-wait_ARQ "Stop-and-wait ARQ"), [Go-Back-N ARQ](https://en.wikipedia.org/wiki/Go-Back-N_ARQ "Go-Back-N ARQ"), and [Selective Repeat ARQ](https://en.wikipedia.org/wiki/Selective_Repeat_ARQ "Selective Repeat ARQ"). ARQ is appropriate if the communication channel has varying or unknown [capacity](https://en.wikipedia.org/wiki/Channel_capacity "Channel capacity"), such as is the case on the Internet. However, ARQ requires the availability of a [back channel](https://en.wikipedia.org/wiki/Backward_channel "Backward channel"), results in possibly increased [latency](https://en.wikipedia.org/wiki/Network_latency "Network latency") due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of [network congestion](https://en.wikipedia.org/wiki/Network_congestion "Network congestion") can put a strain on the server and overall network capacity.[\[10\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-reliable-erasure-code-10) For example, ARQ is used on shortwave radio data links in the form of [ARQ-E](https://en.wikipedia.org/wiki/ARQ-E "ARQ-E"), or combined with multiplexing as [ARQ-M](https://en.wikipedia.org/wiki/ARQ-M "ARQ-M"). ### Forward error correction \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=6 "Edit section: Forward error correction")\] [Forward error correction](https://en.wikipedia.org/wiki/Forward_error_correction "Forward error correction") (FEC) is a process of adding [redundant data](https://en.wikipedia.org/wiki/Redundancy_\(information_theory\) "Redundancy (information theory)") such as an [error-correcting code](https://en.wikipedia.org/wiki/Error-correcting_code "Error-correcting code") (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a [backchannel](https://en.wikipedia.org/wiki/Backchannel "Backchannel") is not required in forward error correction. Error-correcting codes are used in [lower-layer](https://en.wikipedia.org/wiki/Physical_layer "Physical layer") communication such as [cellular network](https://en.wikipedia.org/wiki/Cellular_network "Cellular network"), high-speed [fiber-optic communication](https://en.wikipedia.org/wiki/Fiber-optic_communication "Fiber-optic communication") and [Wi-Fi](https://en.wikipedia.org/wiki/Wi-Fi "Wi-Fi"),[\[11\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-11)[\[12\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-12) as well as for reliable storage in media such as [flash memory](https://en.wikipedia.org/wiki/Flash_memory "Flash memory"), [hard disk](https://en.wikipedia.org/wiki/Hard_disk "Hard disk") and [RAM](https://en.wikipedia.org/wiki/ECC_memory "ECC memory").[\[13\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-13) Error-correcting codes are usually distinguished between [convolutional codes](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code") and [block codes](https://en.wikipedia.org/wiki/Block_code "Block code"): - Convolutional codes are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the [Viterbi decoder](https://en.wikipedia.org/wiki/Viterbi_decoder "Viterbi decoder") allows [optimal decoding](https://en.wikipedia.org/wiki/Maximum_likelihood_decoding "Maximum likelihood decoding"). - Block codes are processed on a [block-by-block](https://en.wikipedia.org/wiki/Block_\(telecommunications\) "Block (telecommunications)") basis. Early examples of block codes are [repetition codes](https://en.wikipedia.org/wiki/Repetition_code "Repetition code"), [Hamming codes](https://en.wikipedia.org/wiki/Hamming_code "Hamming code") and [multidimensional parity-check codes](https://en.wikipedia.org/wiki/Multidimensional_parity-check_code "Multidimensional parity-check code"). They were followed by a number of efficient codes, [Reed–Solomon codes](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_code "Reed–Solomon code") being the most notable due to their current widespread use. [Turbo codes](https://en.wikipedia.org/wiki/Turbo_code "Turbo code") and [low-density parity-check codes](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code") (LDPC) are relatively new constructions that can provide almost [optimal efficiency](https://en.wikipedia.org/wiki/Category:Capacity-approaching_codes "Category:Capacity-approaching codes"). [Shannon's theorem](https://en.wikipedia.org/wiki/Shannon%27s_theorem "Shannon's theorem") is an important theorem in forward error correction, and describes the maximum [information rate](https://en.wikipedia.org/wiki/Information_rate "Information rate") at which reliable communication is possible over a channel that has a certain error probability or [signal-to-noise ratio](https://en.wikipedia.org/wiki/Signal-to-noise_ratio "Signal-to-noise ratio") (SNR). This strict upper limit is expressed in terms of the [channel capacity](https://en.wikipedia.org/wiki/Channel_capacity "Channel capacity"). More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a [discrete memoryless channel](https://en.wikipedia.org/wiki/Channel_model "Channel model") can be made arbitrarily small, provided that the [code rate](https://en.wikipedia.org/wiki/Code_rate "Code rate") is smaller than the channel capacity. The code rate is defined as the fraction k/n of k source symbols and n encoded symbols. The actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have [efficient](https://en.wikipedia.org/wiki/Polynomial_time "Polynomial time") encoding and decoding algorithms. ### Hybrid schemes \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=7 "Edit section: Hybrid schemes")\] Main article: [Hybrid ARQ](https://en.wikipedia.org/wiki/Hybrid_ARQ "Hybrid ARQ") [Hybrid ARQ](https://en.wikipedia.org/wiki/Hybrid_ARQ "Hybrid ARQ") is a combination of ARQ and forward error correction. There are two basic approaches:[\[10\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-reliable-erasure-code-10) - Messages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check). - Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message. The latter approach is particularly attractive on an [erasure channel](https://en.wikipedia.org/wiki/Erasure_channel "Erasure channel") when using a [rateless erasure code](https://en.wikipedia.org/wiki/Rateless_erasure_code "Rateless erasure code"). ## Types of error detection \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=8 "Edit section: Types of error detection")\] Error detection is most commonly realized using a suitable [hash function](https://en.wikipedia.org/wiki/Hash_function "Hash function") (or specifically, a [checksum](https://en.wikipedia.org/wiki/Checksum "Checksum"), [cyclic redundancy check](https://en.wikipedia.org/wiki/Cyclic_redundancy_check "Cyclic redundancy check") or other algorithm). A hash function adds a fixed-length tag to a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided. There exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check's performance in detecting [burst errors](https://en.wikipedia.org/wiki/Burst_error "Burst error")). ### Minimum distance coding \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=9 "Edit section: Minimum distance coding")\] A random-error-correcting code based on [minimum distance coding](https://en.wikipedia.org/wiki/Minimum_distance_coding "Minimum distance coding") can provide a strict guarantee on the number of detectable errors, but it may not protect against a [preimage attack](https://en.wikipedia.org/wiki/Preimage_attack "Preimage attack"). ### Repetition codes \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=10 "Edit section: Repetition codes")\] A [repetition code](https://en.wikipedia.org/wiki/Repetition_code "Repetition code") is a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern `1011`, the four-bit block can be repeated three times, thus producing `1011 1011 1011`. If this twelve-bit pattern was received as `1010 1011 1011` – where the first block is unlike the other two – an error has occurred. A repetition code is very inefficient and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g., `1010 1010 1010` in the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of [numbers stations](https://en.wikipedia.org/wiki/Numbers_station "Numbers station").[\[14\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-15) ### Parity bit \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=11 "Edit section: Parity bit")\] Main article: [Parity bit](https://en.wikipedia.org/wiki/Parity_bit "Parity bit") A parity bit is a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous. Parity bits added to each word sent are called [transverse redundancy checks](https://en.wikipedia.org/wiki/Transverse_redundancy_check "Transverse redundancy check"), while those added at the end of a stream of words are called [longitudinal redundancy checks](https://en.wikipedia.org/wiki/Longitudinal_redundancy_check "Longitudinal redundancy check"). For example, if each of a series of m-bit words has a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words. With more error correction bits, more errors can be detected and in some cases corrected. There are also other bit-grouping techniques. ### Checksum \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=12 "Edit section: Checksum")\] Main article: [Checksum](https://en.wikipedia.org/wiki/Checksum "Checksum") A checksum of a message is a [modular arithmetic](https://en.wikipedia.org/wiki/Modular_arithmetic "Modular arithmetic") sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a [ones'-complement](https://en.wikipedia.org/wiki/Ones%27-complement "Ones'-complement") operation prior to transmission to detect unintentional all-zero messages. Checksum schemes include parity bits, [check digits](https://en.wikipedia.org/wiki/Check_digit "Check digit"), and [longitudinal redundancy checks](https://en.wikipedia.org/wiki/Longitudinal_redundancy_check "Longitudinal redundancy check"). Some checksum schemes, such as the [Damm algorithm](https://en.wikipedia.org/wiki/Damm_algorithm "Damm algorithm"), the [Luhn algorithm](https://en.wikipedia.org/wiki/Luhn_algorithm "Luhn algorithm"), and the [Verhoeff algorithm](https://en.wikipedia.org/wiki/Verhoeff_algorithm "Verhoeff algorithm"), are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers. ### Cyclic redundancy check \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=13 "Edit section: Cyclic redundancy check")\] Main article: [Cyclic redundancy check](https://en.wikipedia.org/wiki/Cyclic_redundancy_check "Cyclic redundancy check") A cyclic redundancy check (CRC) is a non-secure [hash function](https://en.wikipedia.org/wiki/Hash_function "Hash function") designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a generator polynomial, which is used as the [divisor](https://en.wikipedia.org/wiki/Divisor "Divisor") in a [polynomial long division](https://en.wikipedia.org/wiki/Polynomial_long_division "Polynomial long division") over a [finite field](https://en.wikipedia.org/wiki/Finite_field "Finite field"), taking the input data as the [dividend](https://en.wikipedia.org/wiki/Dividend "Dividend"). The [remainder](https://en.wikipedia.org/wiki/Remainder "Remainder") becomes the result. A CRC has properties that make it well suited for detecting [burst errors](https://en.wikipedia.org/wiki/Burst_error "Burst error"). CRCs are particularly easy to implement in hardware and are therefore commonly used in [computer networks](https://en.wikipedia.org/wiki/Computer_network "Computer network") and storage devices such as [hard disk drives](https://en.wikipedia.org/wiki/Hard_disk_drives "Hard disk drives"). The parity bit can be seen as a special-case 1-bit CRC. ### Cryptographic hash function \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=14 "Edit section: Cryptographic hash function")\] Main article: [Cryptographic hash function](https://en.wikipedia.org/wiki/Cryptographic_hash_function "Cryptographic hash function") The output of a cryptographic hash function, also known as a message digest, can provide strong assurances about [data integrity](https://en.wikipedia.org/wiki/Data_integrity "Data integrity"), whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a keyed hash or [message authentication code](https://en.wikipedia.org/wiki/Message_authentication_code "Message authentication code") (MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message. ### Digital signature \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=15 "Edit section: Digital signature")\] Main article: [Digital signature](https://en.wikipedia.org/wiki/Digital_signature "Digital signature") Digital signatures can provide strong assurances about data integrity, whether the changes of the data are accidental or maliciously introduced. Digital signatures are perhaps most notable for being part of the HTTPS protocol for securely browsing the web. ### Error correction code \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=16 "Edit section: Error correction code")\] Main article: [Error correction code](https://en.wikipedia.org/wiki/Error_correction_code "Error correction code") Any error-correcting code can be used for error detection. A code with minimum [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance "Hamming distance"), d, can detect up to d − 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired. Codes with minimum Hamming distance d = 2 are degenerate cases of error-correcting codes and can be used to detect single errors. The parity bit is an example of a single-error-detecting code. ## Applications \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=17 "Edit section: Applications")\] Applications that require low latency (such as telephone conversations) cannot use [automatic repeat request](https://en.wikipedia.org/wiki/Automatic_repeat_request "Automatic repeat request") (ARQ); they must use [forward error correction](https://en.wikipedia.org/wiki/Forward_error_correction "Forward error correction") (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be usable. Applications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available. Applications that use ARQ must have a [return channel](https://en.wikipedia.org/wiki/Return_channel "Return channel"); applications having no return channel cannot use ARQ. Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC. Reliability and inspection engineering also make use of the theory of error-correcting codes,[\[16\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-16) as well as natural language.[\[17\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-DOI10.7275/bjvb-2n37-17) ### Internet \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=18 "Edit section: Internet")\] In a typical [TCP/IP](https://en.wikipedia.org/wiki/TCP/IP "TCP/IP") stack, error control is performed at multiple levels: - Each [Ethernet frame](https://en.wikipedia.org/wiki/Ethernet_frame "Ethernet frame") uses [CRC-32](https://en.wikipedia.org/wiki/CRC-32 "CRC-32") error detection. Frames with detected errors are discarded by the receiver hardware. - The [IPv4](https://en.wikipedia.org/wiki/IPv4 "IPv4") header contains a [checksum](https://en.wikipedia.org/wiki/IPv4_header_checksum "IPv4 header checksum") protecting the contents of the header. [Packets](https://en.wikipedia.org/wiki/Network_packet "Network packet") with incorrect checksums are dropped within the network or at the receiver. - The checksum was omitted from the [IPv6](https://en.wikipedia.org/wiki/IPv6 "IPv6") header in order to minimize processing costs in [network routing](https://en.wikipedia.org/wiki/Network_routing "Network routing") and because current [link layer](https://en.wikipedia.org/wiki/Link_layer "Link layer") technology is assumed to provide sufficient error detection (see also RFC 3819). - [UDP](https://en.wikipedia.org/wiki/User_Datagram_Protocol "User Datagram Protocol") has an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the [network stack](https://en.wikipedia.org/wiki/Network_stack "Network stack"). The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection. - [TCP](https://en.wikipedia.org/wiki/Transmission_Control_Protocol "Transmission Control Protocol") provides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack and eventually get retransmitted using ARQ, either explicitly (such as through [three-way handshake](https://en.wikipedia.org/wiki/Three-way_handshake "Three-way handshake")) or implicitly due to a [timeout](https://en.wikipedia.org/wiki/Timeout_\(computing\) "Timeout (computing)"). ### Deep-space telecommunications \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=19 "Edit section: Deep-space telecommunications")\] The development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded) [convolutional codes](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code") and [Reed–Muller codes](https://en.wikipedia.org/wiki/Reed%E2%80%93Muller_code "Reed–Muller code").[\[18\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-deep-space-codes-18) The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a [bell curve](https://en.wikipedia.org/wiki/Gaussian_function "Gaussian function")), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977. The [Voyager 1](https://en.wikipedia.org/wiki/Voyager_1 "Voyager 1") and [Voyager 2](https://en.wikipedia.org/wiki/Voyager_2 "Voyager 2") missions, which started in 1977, were designed to deliver color imaging and scientific information from [Jupiter](https://en.wikipedia.org/wiki/Jupiter "Jupiter") and [Saturn](https://en.wikipedia.org/wiki/Saturn "Saturn").[\[19\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-voyager-19) This resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally [Viterbi-decoded](https://en.wikipedia.org/wiki/Viterbi_decoder "Viterbi decoder")) convolutional codes that could be [concatenated](https://en.wikipedia.org/wiki/Concatenated_code "Concatenated code") with an outer [Golay (24,12,8) code](https://en.wikipedia.org/wiki/Binary_Golay_code "Binary Golay code"). The Voyager 2 craft additionally supported an implementation of a [Reed–Solomon code](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_code "Reed–Solomon code"). The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft's extended journey to [Uranus](https://en.wikipedia.org/wiki/Uranus "Uranus") and [Neptune](https://en.wikipedia.org/wiki/Neptune "Neptune"). After ECC system upgrades in 1989, both crafts used V2 RSV coding. The [Consultative Committee for Space Data Systems](https://en.wikipedia.org/wiki/Consultative_Committee_for_Space_Data_Systems "Consultative Committee for Space Data Systems") currently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as [Turbo codes](https://en.wikipedia.org/wiki/Turbo_code "Turbo code") or [LDPC codes](https://en.wikipedia.org/wiki/LDPC_code "LDPC code"). The different kinds of deep space and orbital missions that are conducted suggest that trying to find a one-size-fits-all error correction system will be an ongoing problem. For missions close to Earth, the nature of the [noise](https://en.wikipedia.org/wiki/Noise_\(electronics\) "Noise (electronics)") in the [communication channel](https://en.wikipedia.org/wiki/Communication_channel "Communication channel") is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult. ### Satellite broadcasting \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=20 "Edit section: Satellite broadcasting")\] The demand for satellite [transponder](https://en.wikipedia.org/wiki/Transponder "Transponder") bandwidth continues to grow, fueled by the desire to deliver television (including new channels and [high-definition television](https://en.wikipedia.org/wiki/High-definition_television "High-definition television")) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected [modulation](https://en.wikipedia.org/wiki/Modulation "Modulation") scheme and the proportion of capacity consumed by FEC. ### Data storage \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=21 "Edit section: Data storage")\] Error detection and correction codes are often used to improve the reliability of data storage media.[\[20\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-20) A parity track capable of detecting single-bit errors was present on the first [magnetic tape data storage](https://en.wikipedia.org/wiki/Magnetic_tape_data_storage "Magnetic tape data storage") in 1951. The [optimal rectangular code](https://en.wikipedia.org/wiki/Optimal_rectangular_code "Optimal rectangular code") used in [group coded recording](https://en.wikipedia.org/wiki/Group_coded_recording "Group coded recording") tapes not only detects but also corrects single-bit errors. Some [file formats](https://en.wikipedia.org/wiki/File_format "File format"), particularly [archive formats](https://en.wikipedia.org/wiki/Archive_formats "Archive formats"), include a checksum (most often CRC-32\]) to detect corruption and truncation and can employ redundancy or [parity files](https://en.wikipedia.org/wiki/Parity_file "Parity file") to recover portions of corrupted data. [Reed-Solomon codes](https://en.wikipedia.org/wiki/Reed-Solomon_code "Reed-Solomon code") are used in [compact discs](https://en.wikipedia.org/wiki/Compact_disc "Compact disc") to correct errors caused by scratches. Modern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors.[\[21\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-21) [RAID](https://en.wikipedia.org/wiki/RAID "RAID") systems use a variety of error correction techniques to recover data when a hard drive completely fails. Filesystems such as [ZFS](https://en.wikipedia.org/wiki/ZFS "ZFS") or [Btrfs](https://en.wikipedia.org/wiki/Btrfs "Btrfs"), as well as some [RAID](https://en.wikipedia.org/wiki/RAID "RAID") implementations, support [data scrubbing](https://en.wikipedia.org/wiki/Data_scrubbing "Data scrubbing") and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used.[\[22\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-22) The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware. ### Error-correcting memory \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=22 "Edit section: Error-correcting memory")\] Main article: [ECC memory](https://en.wikipedia.org/wiki/ECC_memory "ECC memory") [Dynamic random-access memory](https://en.wikipedia.org/wiki/Dynamic_random-access_memory "Dynamic random-access memory") (DRAM) may provide stronger protection against [soft errors](https://en.wikipedia.org/wiki/Soft_error "Soft error") by relying on error-correcting codes. Such error-correcting memory, known as ECC or EDAC-protected memory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased [radiation](https://en.wikipedia.org/wiki/Cosmic_ray "Cosmic ray") in space. Error-correcting memory controllers traditionally use [Hamming codes](https://en.wikipedia.org/wiki/Hamming_code "Hamming code"), although some use [triple modular redundancy](https://en.wikipedia.org/wiki/Triple_modular_redundancy "Triple modular redundancy"). [Interleaving](https://en.wikipedia.org/wiki/Error_correction_code#Interleaving "Error correction code") allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a [single-event upset](https://en.wikipedia.org/wiki/Single-event_upset "Single-event upset") (SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained.[\[23\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-23) In addition to hardware providing features required for ECC memory to operate, [operating systems](https://en.wikipedia.org/wiki/Operating_system "Operating system") usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the [Linux kernel](https://en.wikipedia.org/wiki/Linux_kernel "Linux kernel")'s EDAC subsystem (previously known as Bluesmoke), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the [PCI bus](https://en.wikipedia.org/wiki/PCI_bus "PCI bus").[\[24\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-24)[\[25\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-25)[\[26\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-26) A few systems\[[specify](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources "Wikipedia:Citing sources")\] also support [memory scrubbing](https://en.wikipedia.org/wiki/Memory_scrubbing "Memory scrubbing") to catch and correct errors early before they become unrecoverable. ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=23 "Edit section: See also")\] - [Berger code](https://en.wikipedia.org/wiki/Berger_code "Berger code") - [Burst error-correcting code](https://en.wikipedia.org/wiki/Burst_error-correcting_code "Burst error-correcting code") - [ECC memory](https://en.wikipedia.org/wiki/ECC_memory "ECC memory"), a type of computer data storage - [Link adaptation](https://en.wikipedia.org/wiki/Link_adaptation "Link adaptation") - [List of algorithms § Error detection and correction](https://en.wikipedia.org/wiki/List_of_algorithms#Error_detection_and_correction "List of algorithms") - [List of hash functions](https://en.wikipedia.org/wiki/List_of_hash_functions "List of hash functions") ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=24 "Edit section: References")\] 1. ^ [*a](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-Jewish_1-0) [b](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-Jewish_1-1) "Masorah". [Jewish Encyclopedia](http://www.jewishencyclopedia.com/articles/10465-masorah). 2. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-2)* Pratico, Gary D.; Pelt, Miles V. Van (2009). [Basics of Biblical Hebrew Grammar: Second Edition](https://books.google.com/books?id=p3FIoT0s3yYC&pg=PT448). Zondervan. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-310-55882-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-310-55882-8 "Special:BookSources/978-0-310-55882-8") . 3. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-3) Mounce, William D. (2007). [Greek for the Rest of Us: Using Greek Tools Without Mastering Biblical Languages](https://books.google.com/books?id=AF-5ptJ0l2gC&pg=PA289). Zondervan. p. 289. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-310-28289-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-310-28289-1 "Special:BookSources/978-0-310-28289-1") . 4. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-4) [Mishneh Torah](https://en.wikipedia.org/wiki/Mishneh_Torah "Mishneh Torah"), Tefillin, Mezuzah, and Sefer Torah, 1:2. Example English translation: [Eliyahu Touger](https://en.wikipedia.org/w/index.php?title=Eliyahu_Touger&action=edit&redlink=1 "Eliyahu Touger (page does not exist)"). [The Rambam's Mishneh Torah](https://www.chabad.org/library/article_cdo/aid/925417/jewish/Tefillin-Mezuzah-and-Sefer-Torah-Chapter-One.htm). [Moznaim Publishing Corporation](https://en.wikipedia.org/w/index.php?title=Moznaim_Publishing_Corporation&action=edit&redlink=1 "Moznaim Publishing Corporation (page does not exist)"). 5. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-5) Brian M. Fagan (5 December 1996). "Dead Sea Scrolls". [The Oxford Companion to Archaeology](https://en.wikipedia.org/w/index.php?title=The_Oxford_Companion_to_Archaeology&action=edit&redlink=1 "The Oxford Companion to Archaeology (page does not exist)"). [Oxford University Press](https://en.wikipedia.org/wiki/Oxford_University_Press "Oxford University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0195076184](https://en.wikipedia.org/wiki/Special:BookSources/0195076184 "Special:BookSources/0195076184") . 6. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-Thompson_6-0) Thompson, Thomas M. (1983), From Error-Correcting Codes through Sphere Packings to Simple Groups, The Carus Mathematical Monographs (\#21), The Mathematical Association of America, p. vii, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-88385-023-0](https://en.wikipedia.org/wiki/Special:BookSources/0-88385-023-0 "Special:BookSources/0-88385-023-0") 7. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-7) Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal, 27 (3): 379–423, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/j.1538-7305.1948.tb01338.x](https://doi.org/10.1002%2Fj.1538-7305.1948.tb01338.x), [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10338\.dmlcz/101429](https://hdl.handle.net/10338.dmlcz%2F101429), [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [9230594](https://pubmed.ncbi.nlm.nih.gov/9230594) 8. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-8) Golay, Marcel J. E. (1949), "Notes on Digital Coding", Proc.I.R.E. (I.E.E.E.), 37: 657 9. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-9) Gupta, Vikas; Verma, Chanderkant (November 2012). "Error Detection and Correction: An Introduction". International Journal of Advanced Research in Computer Science and Software Engineering. 2 (11). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [17499858](https://api.semanticscholar.org/CorpusID:17499858). 10. ^ [*a](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-reliable-erasure-code_10-0) [b](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-reliable-erasure-code_10-1) A. J. McAuley, Reliable Broadband Communication Using a Burst Erasure Correcting Code, ACM SIGCOMM, 1990. 11. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-11)* Shah, Pradeep M.; Vyavahare, Prakash D.; Jain, Anjana (September 2015). ["Modern error correcting codes for 4G and beyond: Turbo codes and LDPC codes"](https://www.researchgate.net/publication/301611980). 2015 Radio and Antenna Days of the Indian Ocean (RADIO). pp. 1–2\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/RADIO.2015.7323369](https://doi.org/10.1109%2FRADIO.2015.7323369). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-9-9903-7339-4](https://en.wikipedia.org/wiki/Special:BookSources/978-9-9903-7339-4 "Special:BookSources/978-9-9903-7339-4") . [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [28885076](https://api.semanticscholar.org/CorpusID:28885076). Retrieved 22 May 2022. 12. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-12) ["IEEE SA - IEEE 802.11ac-2013"](https://web.archive.org/web/20220522205452/https://standards.ieee.org/ieee/802.11ac/4473/). IEEE Standards Association. Archived from [the original](https://standards.ieee.org/ieee/802.11ac/4473/) on 2022-05-22. Retrieved 2022-05-22. 13. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-13) ["Transition to Advanced Format 4K Sector Hard Drives \\| Seagate US"](https://www.seagate.com/sg/en/tech-insights/advanced-format-4k-sector-hard-drives-master-ti/). Seagate.com. Retrieved 22 May 2022. 14. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-14) Frank van Gerwen. ["Numbers (and other mysterious) stations"](https://web.archive.org/web/20170712053335/http://www.cisquet.nl/numbers.htm). Archived from [the original](http://www.cisquet.nl/numbers.htm) on 12 July 2017. Retrieved 12 March 2012. 15. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-15) Gary Cutlack (25 August 2010). ["Mysterious Russian 'Numbers Station' Changes Broadcast After 20 Years"](https://web.archive.org/web/20170705131816/https://www.gizmodo.com.au/2010/08/mysterious-russian-numbers-station-changes-broadcast-after-20-years/). Gizmodo. Archived from [the original](http://www.gizmodo.com.au/2010/08/mysterious-russian-numbers-station-changes-broadcast-after-20-years/) on 5 July 2017. Retrieved 12 March 2012. 16. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-16) Ben-Gal I.; Herer Y.; Raz T. (2003). ["Self-correcting inspection procedure under inspection errors"](https://web.archive.org/web/20131013171945/http://www.eng.tau.ac.il/~bengal/SCI_paper.pdf) (PDF). IIE Transactions. IIE Transactions on Quality and Reliability, 34(6), pp. 529-540. Archived from [the original](http://www.eng.tau.ac.il/~bengal/SCI_paper.pdf) (PDF) on 2013-10-13. Retrieved 2014-01-10. 17. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-DOI10.7275/bjvb-2n37_17-0) Yvo Meeres, Tommi A. Pirinen (2021). "Vowel Harmony Viewed as Error-Correcting Code". Proceedings of the Society for Computation in Linguistics. 4 (1): 313–322\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.7275/bjvb-2n37](https://doi.org/10.7275%2Fbjvb-2n37). 18. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-deep-space-codes_18-0) K. Andrews et al., The Development of Turbo and LDPC Codes for Deep-Space Applications, Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007. 19. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-voyager_19-0) Huffman, William Cary; [Pless, Vera S.](https://en.wikipedia.org/wiki/Vera_Pless "Vera Pless") (2003). [Fundamentals of Error-Correcting Codes](https://archive.org/details/fundamentalsofer0000huff). [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-521-78280-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-78280-7 "Special:BookSources/978-0-521-78280-7") . 20. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-20) Kurtas, Erozan M.; Vasic, Bane (2018-10-03). Advanced Error Control Techniques for Data Storage Systems. CRC Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4200-3649-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4200-3649-7 "Special:BookSources/978-1-4200-3649-7") . 21. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-21) Scott A. Moulton. ["My Hard Drive Died"](https://web.archive.org/web/20080202143103/http://www.myharddrivedied.com/presentations_whitepaper.html). Archived from [the original](http://www.myharddrivedied.com/presentations_whitepaper.html) on 2008-02-02. 22. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-22) Qiao, Zhi; Fu, Song; Chen, Hsing-Bung; Settlemyer, Bradley (2019). "Building Reliable High-Performance Storage Systems: An Empirical and Analytical Study". 2019 IEEE International Conference on Cluster Computing (CLUSTER). pp. 1–10\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/CLUSTER.2019.8891006](https://doi.org/10.1109%2FCLUSTER.2019.8891006). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-7281-4734-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-7281-4734-5 "Special:BookSources/978-1-7281-4734-5") . [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [207951690](https://api.semanticscholar.org/CorpusID:207951690). 23. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-23) ["Using StrongArm SA-1110 in the On-Board Computer of Nanosatellite"](https://web.archive.org/web/20111002152735/http://www.apmcsta.org/File/doc/Conferences/6th%20meeting/Chen%20Zhenyu.doc). Tsinghua Space Center, [Tsinghua University](https://en.wikipedia.org/wiki/Tsinghua_University "Tsinghua University"), Beijing. Archived from [the original](http://www.apmcsta.org/File/doc/Conferences/6th%20meeting/Chen%20Zhenyu.doc) on 2011-10-02. Retrieved 2009-02-16. 24. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-24) Jeff Layton. ["Error Detection and Correction"](http://www.admin-magazine.com/Articles/Monitoring-Memory-Errors). [Linux Magazine](https://en.wikipedia.org/wiki/Linux_Magazine "Linux Magazine"). Retrieved 2014-08-12. 25. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-25) ["EDAC Project"](https://bluesmoke.sourceforge.net/). bluesmoke.sourceforge.net. Retrieved 2014-08-12. 26. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-26) ["Documentation/edac.txt"](https://web.archive.org/web/20090905174616/http://www.kernel.org/doc/Documentation/edac.txt). Linux kernel documentation. [kernel.org](https://en.wikipedia.org/wiki/Kernel.org "Kernel.org"). 2014-06-16. Archived from [the original](https://www.kernel.org/doc/Documentation/edac.txt) on 2009-09-05. Retrieved 2014-08-12. ## Further reading \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=25 "Edit section: Further reading")\] - Shu Lin; Daniel J. Costello, Jr. (1983). Error Control Coding: Fundamentals and Applications. [Prentice Hall](https://en.wikipedia.org/wiki/Prentice_Hall "Prentice Hall"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-13-283796-X](https://en.wikipedia.org/wiki/Special:BookSources/0-13-283796-X "Special:BookSources/0-13-283796-X") . - [SoftECC: A System for Software Memory Integrity Checking](http://pdos.csail.mit.edu/papers/softecc:ddopson-meng/softecc_ddopson-meng.pdf) - [A Tunable, Software-based DRAM Error Detection and Correction Library for HPC](http://www.fiala.me/pubs/papers/libsdc11.pdf) [Archived](https://web.archive.org/web/20141107074502/http://www.fiala.me/pubs/papers/libsdc11.pdf) 2014-11-07 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") - [Detection and Correction of Silent Data Corruption for Large-Scale High-Performance Computing](http://www.fiala.me/pubs/papers/sc12-redmpi.pdf) [Archived](https://web.archive.org/web/20141107074511/http://www.fiala.me/pubs/papers/sc12-redmpi.pdf) 2014-11-07 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=26 "Edit section: External links")\] - [The on-line textbook: Information Theory, Inference, and Learning Algorithms](http://www.inference.phy.cam.ac.uk/mackay/itila/), by [David J.C. MacKay](https://en.wikipedia.org/wiki/David_J.C._MacKay "David J.C. MacKay"), contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including [low-density parity-check codes](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code"), [turbo codes](https://en.wikipedia.org/wiki/Turbo_code "Turbo code"), and [fountain codes](https://en.wikipedia.org/wiki/Fountain_codes "Fountain codes"). - [ECC Page](http://www.eccpage.com/) – implementations of popular ECC encoding and decoding routines ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1) Retrieved from "<https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&oldid=1337702567>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - [Error detection and correction](https://en.wikipedia.org/wiki/Category:Error_detection_and_correction "Category:Error detection and correction") - [Computer errors](https://en.wikipedia.org/wiki/Category:Computer_errors "Category:Computer errors") Hidden categories: - [Articles with short description](https://en.wikipedia.org/wiki/Category:Articles_with_short_description "Category:Articles with short description") - [Short description is different from Wikidata](https://en.wikipedia.org/wiki/Category:Short_description_is_different_from_Wikidata "Category:Short description is different from Wikidata") - [Articles needing additional references from August 2008](https://en.wikipedia.org/wiki/Category:Articles_needing_additional_references_from_August_2008 "Category:Articles needing additional references from August 2008") - [All articles needing additional references](https://en.wikipedia.org/wiki/Category:All_articles_needing_additional_references "Category:All articles needing additional references") - [Articles needing more detailed references](https://en.wikipedia.org/wiki/Category:Articles_needing_more_detailed_references "Category:Articles needing more detailed references") - [Webarchive template wayback links](https://en.wikipedia.org/wiki/Category:Webarchive_template_wayback_links "Category:Webarchive template wayback links") - This page was last edited on 11 February 2026, at 00:30 (UTC). - Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License "Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License"); additional terms may apply. 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Readable Markdown	[![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg/250px-Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg)](https://en.wikipedia.org/wiki/File:Reed%E2%80%93Solomon_error_correction_Mona_Lisa_LroLrLasercomFig4.jpg) To clean up transmission errors introduced by Earth's atmosphere (left), Goddard scientists applied [Reed–Solomon error correction](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction "Reed–Solomon error correction") (right), which is commonly used in CDs and DVDs. Typical errors include missing pixels (white) and false signals (black). The white stripe indicates a brief period when transmission was interrupted. In [information theory](https://en.wikipedia.org/wiki/Information_theory "Information theory") and [coding theory](https://en.wikipedia.org/wiki/Coding_theory "Coding theory") with applications in [computer science](https://en.wikipedia.org/wiki/Computer_science "Computer science") and [telecommunications](https://en.wikipedia.org/wiki/Telecommunications "Telecommunications"), error detection and correction (EDAC) or error control are techniques that enable [reliable delivery](https://en.wikipedia.org/wiki/Reliable_delivery "Reliable delivery") of [digital data](https://en.wikipedia.org/wiki/Digital_data "Digital data") over unreliable [communication channels](https://en.wikipedia.org/wiki/Communication_channel "Communication channel"). Many communication channels are subject to [channel noise](https://en.wikipedia.org/wiki/Channel_noise "Channel noise"), and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. Error detection is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. Error correction is the detection of errors and reconstruction of the original, error-free data. In classical antiquity, [copyists](https://en.wikipedia.org/wiki/Copyist "Copyist") of the [Hebrew Bible](https://en.wikipedia.org/wiki/Hebrew_Bible "Hebrew Bible") were paid for their work according to the number of [stichs](https://en.wiktionary.org/wiki/stich "wiktionary:stich") (lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters.[\[1\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-Jewish-1) This also helped ensure accuracy in the transmission of the text with the production of subsequent copies.[\[2\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-2)[\[3\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-3) Between the 7th and 10th centuries CE a [group of Jewish scribes](https://en.wikipedia.org/wiki/Masoretes "Masoretes") formalized and expanded this to create the [Numerical Masorah](https://en.wikipedia.org/wiki/Masoretic_Text#Numerical_Masorah "Masoretic Text") to ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary.[\[1\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-Jewish-1) Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable.[\[4\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-4) The effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the [Dead Sea Scrolls](https://en.wikipedia.org/wiki/Dead_Sea_Scrolls "Dead Sea Scrolls") in 1947–1956, dating from c. 150 BCE – 75 CE.[\[5\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-5) The modern development of [error correction codes](https://en.wikipedia.org/wiki/Error_correction_code "Error correction code") is credited to [Richard Hamming](https://en.wikipedia.org/wiki/Richard_Hamming "Richard Hamming") in 1947.[\[6\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-Thompson-6) A description of [Hamming's code](https://en.wikipedia.org/wiki/Hamming%27s_code "Hamming's code") appeared in [Claude Shannon](https://en.wikipedia.org/wiki/Claude_Shannon "Claude Shannon")'s A Mathematical Theory of Communication[\[7\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-7) and was quickly generalized by [Marcel J. E. Golay](https://en.wikipedia.org/wiki/Marcel_J._E._Golay "Marcel J. E. Golay").[\[8\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-8) All error-detection and correction schemes add some [redundancy](https://en.wikipedia.org/wiki/Redundancy_\(information_theory\) "Redundancy (information theory)") (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted. Error detection and correction schemes can be either [systematic](https://en.wikipedia.org/wiki/Systematic_code "Systematic code") or non-systematic. In a systematic scheme, the transmitter sends the original (error-free) data and attaches a fixed number of check bits (or parity data), which are derived from the data bits by some encoding algorithm. If error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message. Good error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common [channel models](https://en.wikipedia.org/wiki/Channel_model "Channel model") include [memoryless](https://en.wikipedia.org/wiki/Memoryless "Memoryless") models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in [bursts](https://en.wikipedia.org/wiki/Burst_error "Burst error"). Consequently, error-detecting and -correcting codes can be generally distinguished between random-error-detecting/correcting and burst-error-detecting/correcting. Some codes can also be suitable for a mixture of random errors and burst errors. If the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as [automatic repeat request](https://en.wikipedia.org/wiki/Automatic_repeat_request "Automatic repeat request") (ARQ), and is most notably used in the Internet. An alternate approach for error control is [hybrid automatic repeat request](https://en.wikipedia.org/wiki/Hybrid_automatic_repeat_request "Hybrid automatic repeat request") (HARQ), which is a combination of ARQ and error-correction coding. ## Types of error correction \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=4 "Edit section: Types of error correction")\] There are three major types of error correction:[\[9\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-9) ### Automatic repeat request \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=5 "Edit section: Automatic repeat request")\] [Automatic repeat request](https://en.wikipedia.org/wiki/Automatic_repeat_request "Automatic repeat request") (ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and/or negative acknowledgment messages, and [timeouts](https://en.wikipedia.org/wiki/Timeout_\(computing\) "Timeout (computing)") to achieve reliable data transmission. An acknowledgment is a message sent by the receiver to indicate that it has correctly received a [data frame](https://en.wikipedia.org/wiki/Data_frame "Data frame"). Usually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e., within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions. Three types of ARQ protocols are [Stop-and-wait ARQ](https://en.wikipedia.org/wiki/Stop-and-wait_ARQ "Stop-and-wait ARQ"), [Go-Back-N ARQ](https://en.wikipedia.org/wiki/Go-Back-N_ARQ "Go-Back-N ARQ"), and [Selective Repeat ARQ](https://en.wikipedia.org/wiki/Selective_Repeat_ARQ "Selective Repeat ARQ"). ARQ is appropriate if the communication channel has varying or unknown [capacity](https://en.wikipedia.org/wiki/Channel_capacity "Channel capacity"), such as is the case on the Internet. However, ARQ requires the availability of a [back channel](https://en.wikipedia.org/wiki/Backward_channel "Backward channel"), results in possibly increased [latency](https://en.wikipedia.org/wiki/Network_latency "Network latency") due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of [network congestion](https://en.wikipedia.org/wiki/Network_congestion "Network congestion") can put a strain on the server and overall network capacity.[\[10\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-reliable-erasure-code-10) For example, ARQ is used on shortwave radio data links in the form of [ARQ-E](https://en.wikipedia.org/wiki/ARQ-E "ARQ-E"), or combined with multiplexing as [ARQ-M](https://en.wikipedia.org/wiki/ARQ-M "ARQ-M"). ### Forward error correction \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=6 "Edit section: Forward error correction")\] [Forward error correction](https://en.wikipedia.org/wiki/Forward_error_correction "Forward error correction") (FEC) is a process of adding [redundant data](https://en.wikipedia.org/wiki/Redundancy_\(information_theory\) "Redundancy (information theory)") such as an [error-correcting code](https://en.wikipedia.org/wiki/Error-correcting_code "Error-correcting code") (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a [backchannel](https://en.wikipedia.org/wiki/Backchannel "Backchannel") is not required in forward error correction. Error-correcting codes are used in [lower-layer](https://en.wikipedia.org/wiki/Physical_layer "Physical layer") communication such as [cellular network](https://en.wikipedia.org/wiki/Cellular_network "Cellular network"), high-speed [fiber-optic communication](https://en.wikipedia.org/wiki/Fiber-optic_communication "Fiber-optic communication") and [Wi-Fi](https://en.wikipedia.org/wiki/Wi-Fi "Wi-Fi"),[\[11\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-11)[\[12\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-12) as well as for reliable storage in media such as [flash memory](https://en.wikipedia.org/wiki/Flash_memory "Flash memory"), [hard disk](https://en.wikipedia.org/wiki/Hard_disk "Hard disk") and [RAM](https://en.wikipedia.org/wiki/ECC_memory "ECC memory").[\[13\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-13) Error-correcting codes are usually distinguished between [convolutional codes](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code") and [block codes](https://en.wikipedia.org/wiki/Block_code "Block code"): - Convolutional codes are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the [Viterbi decoder](https://en.wikipedia.org/wiki/Viterbi_decoder "Viterbi decoder") allows [optimal decoding](https://en.wikipedia.org/wiki/Maximum_likelihood_decoding "Maximum likelihood decoding"). - Block codes are processed on a [block-by-block](https://en.wikipedia.org/wiki/Block_\(telecommunications\) "Block (telecommunications)") basis. Early examples of block codes are [repetition codes](https://en.wikipedia.org/wiki/Repetition_code "Repetition code"), [Hamming codes](https://en.wikipedia.org/wiki/Hamming_code "Hamming code") and [multidimensional parity-check codes](https://en.wikipedia.org/wiki/Multidimensional_parity-check_code "Multidimensional parity-check code"). They were followed by a number of efficient codes, [Reed–Solomon codes](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_code "Reed–Solomon code") being the most notable due to their current widespread use. [Turbo codes](https://en.wikipedia.org/wiki/Turbo_code "Turbo code") and [low-density parity-check codes](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code") (LDPC) are relatively new constructions that can provide almost [optimal efficiency](https://en.wikipedia.org/wiki/Category:Capacity-approaching_codes "Category:Capacity-approaching codes"). [Shannon's theorem](https://en.wikipedia.org/wiki/Shannon%27s_theorem "Shannon's theorem") is an important theorem in forward error correction, and describes the maximum [information rate](https://en.wikipedia.org/wiki/Information_rate "Information rate") at which reliable communication is possible over a channel that has a certain error probability or [signal-to-noise ratio](https://en.wikipedia.org/wiki/Signal-to-noise_ratio "Signal-to-noise ratio") (SNR). This strict upper limit is expressed in terms of the [channel capacity](https://en.wikipedia.org/wiki/Channel_capacity "Channel capacity"). More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a [discrete memoryless channel](https://en.wikipedia.org/wiki/Channel_model "Channel model") can be made arbitrarily small, provided that the [code rate](https://en.wikipedia.org/wiki/Code_rate "Code rate") is smaller than the channel capacity. The code rate is defined as the fraction k/n of k source symbols and n encoded symbols. The actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon's proof was only of existential nature, and did not show how to construct codes that are both optimal and have [efficient](https://en.wikipedia.org/wiki/Polynomial_time "Polynomial time") encoding and decoding algorithms. [Hybrid ARQ](https://en.wikipedia.org/wiki/Hybrid_ARQ "Hybrid ARQ") is a combination of ARQ and forward error correction. There are two basic approaches:[\[10\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-reliable-erasure-code-10) - Messages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check). - Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message. The latter approach is particularly attractive on an [erasure channel](https://en.wikipedia.org/wiki/Erasure_channel "Erasure channel") when using a [rateless erasure code](https://en.wikipedia.org/wiki/Rateless_erasure_code "Rateless erasure code"). ## Types of error detection \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=8 "Edit section: Types of error detection")\] Error detection is most commonly realized using a suitable [hash function](https://en.wikipedia.org/wiki/Hash_function "Hash function") (or specifically, a [checksum](https://en.wikipedia.org/wiki/Checksum "Checksum"), [cyclic redundancy check](https://en.wikipedia.org/wiki/Cyclic_redundancy_check "Cyclic redundancy check") or other algorithm). A hash function adds a fixed-length tag to a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided. There exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check's performance in detecting [burst errors](https://en.wikipedia.org/wiki/Burst_error "Burst error")). ### Minimum distance coding \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=9 "Edit section: Minimum distance coding")\] A random-error-correcting code based on [minimum distance coding](https://en.wikipedia.org/wiki/Minimum_distance_coding "Minimum distance coding") can provide a strict guarantee on the number of detectable errors, but it may not protect against a [preimage attack](https://en.wikipedia.org/wiki/Preimage_attack "Preimage attack"). A [repetition code](https://en.wikipedia.org/wiki/Repetition_code "Repetition code") is a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern 1011, the four-bit block can be repeated three times, thus producing 1011 1011 1011. If this twelve-bit pattern was received as 1010 1011 1011 – where the first block is unlike the other two – an error has occurred. A repetition code is very inefficient and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g., 1010 1010 1010 in the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of [numbers stations](https://en.wikipedia.org/wiki/Numbers_station "Numbers station").[\[14\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-15) A parity bit is a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous. Parity bits added to each word sent are called [transverse redundancy checks](https://en.wikipedia.org/wiki/Transverse_redundancy_check "Transverse redundancy check"), while those added at the end of a stream of words are called [longitudinal redundancy checks](https://en.wikipedia.org/wiki/Longitudinal_redundancy_check "Longitudinal redundancy check"). For example, if each of a series of m-bit words has a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words. With more error correction bits, more errors can be detected and in some cases corrected. There are also other bit-grouping techniques. A checksum of a message is a [modular arithmetic](https://en.wikipedia.org/wiki/Modular_arithmetic "Modular arithmetic") sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a [ones'-complement](https://en.wikipedia.org/wiki/Ones%27-complement "Ones'-complement") operation prior to transmission to detect unintentional all-zero messages. Checksum schemes include parity bits, [check digits](https://en.wikipedia.org/wiki/Check_digit "Check digit"), and [longitudinal redundancy checks](https://en.wikipedia.org/wiki/Longitudinal_redundancy_check "Longitudinal redundancy check"). Some checksum schemes, such as the [Damm algorithm](https://en.wikipedia.org/wiki/Damm_algorithm "Damm algorithm"), the [Luhn algorithm](https://en.wikipedia.org/wiki/Luhn_algorithm "Luhn algorithm"), and the [Verhoeff algorithm](https://en.wikipedia.org/wiki/Verhoeff_algorithm "Verhoeff algorithm"), are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers. ### Cyclic redundancy check \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=13 "Edit section: Cyclic redundancy check")\] A cyclic redundancy check (CRC) is a non-secure [hash function](https://en.wikipedia.org/wiki/Hash_function "Hash function") designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a generator polynomial, which is used as the [divisor](https://en.wikipedia.org/wiki/Divisor "Divisor") in a [polynomial long division](https://en.wikipedia.org/wiki/Polynomial_long_division "Polynomial long division") over a [finite field](https://en.wikipedia.org/wiki/Finite_field "Finite field"), taking the input data as the [dividend](https://en.wikipedia.org/wiki/Dividend "Dividend"). The [remainder](https://en.wikipedia.org/wiki/Remainder "Remainder") becomes the result. A CRC has properties that make it well suited for detecting [burst errors](https://en.wikipedia.org/wiki/Burst_error "Burst error"). CRCs are particularly easy to implement in hardware and are therefore commonly used in [computer networks](https://en.wikipedia.org/wiki/Computer_network "Computer network") and storage devices such as [hard disk drives](https://en.wikipedia.org/wiki/Hard_disk_drives "Hard disk drives"). The parity bit can be seen as a special-case 1-bit CRC. ### Cryptographic hash function \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=14 "Edit section: Cryptographic hash function")\] The output of a cryptographic hash function, also known as a message digest, can provide strong assurances about [data integrity](https://en.wikipedia.org/wiki/Data_integrity "Data integrity"), whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a keyed hash or [message authentication code](https://en.wikipedia.org/wiki/Message_authentication_code "Message authentication code") (MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message. Digital signatures can provide strong assurances about data integrity, whether the changes of the data are accidental or maliciously introduced. Digital signatures are perhaps most notable for being part of the HTTPS protocol for securely browsing the web. ### Error correction code \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=16 "Edit section: Error correction code")\] Any error-correcting code can be used for error detection. A code with minimum [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance "Hamming distance"), d, can detect up to d − 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired. Codes with minimum Hamming distance d = 2 are degenerate cases of error-correcting codes and can be used to detect single errors. The parity bit is an example of a single-error-detecting code. Applications that require low latency (such as telephone conversations) cannot use [automatic repeat request](https://en.wikipedia.org/wiki/Automatic_repeat_request "Automatic repeat request") (ARQ); they must use [forward error correction](https://en.wikipedia.org/wiki/Forward_error_correction "Forward error correction") (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be usable. Applications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available. Applications that use ARQ must have a [return channel](https://en.wikipedia.org/wiki/Return_channel "Return channel"); applications having no return channel cannot use ARQ. Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC. Reliability and inspection engineering also make use of the theory of error-correcting codes,[\[16\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-16) as well as natural language.[\[17\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-DOI10.7275/bjvb-2n37-17) In a typical [TCP/IP](https://en.wikipedia.org/wiki/TCP/IP "TCP/IP") stack, error control is performed at multiple levels: - Each [Ethernet frame](https://en.wikipedia.org/wiki/Ethernet_frame "Ethernet frame") uses [CRC-32](https://en.wikipedia.org/wiki/CRC-32 "CRC-32") error detection. Frames with detected errors are discarded by the receiver hardware. - The [IPv4](https://en.wikipedia.org/wiki/IPv4 "IPv4") header contains a [checksum](https://en.wikipedia.org/wiki/IPv4_header_checksum "IPv4 header checksum") protecting the contents of the header. [Packets](https://en.wikipedia.org/wiki/Network_packet "Network packet") with incorrect checksums are dropped within the network or at the receiver. - The checksum was omitted from the [IPv6](https://en.wikipedia.org/wiki/IPv6 "IPv6") header in order to minimize processing costs in [network routing](https://en.wikipedia.org/wiki/Network_routing "Network routing") and because current [link layer](https://en.wikipedia.org/wiki/Link_layer "Link layer") technology is assumed to provide sufficient error detection (see also RFC 3819). - [UDP](https://en.wikipedia.org/wiki/User_Datagram_Protocol "User Datagram Protocol") has an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the [network stack](https://en.wikipedia.org/wiki/Network_stack "Network stack"). The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection. - [TCP](https://en.wikipedia.org/wiki/Transmission_Control_Protocol "Transmission Control Protocol") provides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack and eventually get retransmitted using ARQ, either explicitly (such as through [three-way handshake](https://en.wikipedia.org/wiki/Three-way_handshake "Three-way handshake")) or implicitly due to a [timeout](https://en.wikipedia.org/wiki/Timeout_\(computing\) "Timeout (computing)"). ### Deep-space telecommunications \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=19 "Edit section: Deep-space telecommunications")\] The development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded) [convolutional codes](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code") and [Reed–Muller codes](https://en.wikipedia.org/wiki/Reed%E2%80%93Muller_code "Reed–Muller code").[\[18\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-deep-space-codes-18) The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a [bell curve](https://en.wikipedia.org/wiki/Gaussian_function "Gaussian function")), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977. The [Voyager 1](https://en.wikipedia.org/wiki/Voyager_1 "Voyager 1") and [Voyager 2](https://en.wikipedia.org/wiki/Voyager_2 "Voyager 2") missions, which started in 1977, were designed to deliver color imaging and scientific information from [Jupiter](https://en.wikipedia.org/wiki/Jupiter "Jupiter") and [Saturn](https://en.wikipedia.org/wiki/Saturn "Saturn").[\[19\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-voyager-19) This resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally [Viterbi-decoded](https://en.wikipedia.org/wiki/Viterbi_decoder "Viterbi decoder")) convolutional codes that could be [concatenated](https://en.wikipedia.org/wiki/Concatenated_code "Concatenated code") with an outer [Golay (24,12,8) code](https://en.wikipedia.org/wiki/Binary_Golay_code "Binary Golay code"). The Voyager 2 craft additionally supported an implementation of a [Reed–Solomon code](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_code "Reed–Solomon code"). The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft's extended journey to [Uranus](https://en.wikipedia.org/wiki/Uranus "Uranus") and [Neptune](https://en.wikipedia.org/wiki/Neptune "Neptune"). After ECC system upgrades in 1989, both crafts used V2 RSV coding. The [Consultative Committee for Space Data Systems](https://en.wikipedia.org/wiki/Consultative_Committee_for_Space_Data_Systems "Consultative Committee for Space Data Systems") currently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as [Turbo codes](https://en.wikipedia.org/wiki/Turbo_code "Turbo code") or [LDPC codes](https://en.wikipedia.org/wiki/LDPC_code "LDPC code"). The different kinds of deep space and orbital missions that are conducted suggest that trying to find a one-size-fits-all error correction system will be an ongoing problem. For missions close to Earth, the nature of the [noise](https://en.wikipedia.org/wiki/Noise_\(electronics\) "Noise (electronics)") in the [communication channel](https://en.wikipedia.org/wiki/Communication_channel "Communication channel") is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult. ### Satellite broadcasting \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=20 "Edit section: Satellite broadcasting")\] The demand for satellite [transponder](https://en.wikipedia.org/wiki/Transponder "Transponder") bandwidth continues to grow, fueled by the desire to deliver television (including new channels and [high-definition television](https://en.wikipedia.org/wiki/High-definition_television "High-definition television")) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected [modulation](https://en.wikipedia.org/wiki/Modulation "Modulation") scheme and the proportion of capacity consumed by FEC. Error detection and correction codes are often used to improve the reliability of data storage media.[\[20\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-20) A parity track capable of detecting single-bit errors was present on the first [magnetic tape data storage](https://en.wikipedia.org/wiki/Magnetic_tape_data_storage "Magnetic tape data storage") in 1951. The [optimal rectangular code](https://en.wikipedia.org/wiki/Optimal_rectangular_code "Optimal rectangular code") used in [group coded recording](https://en.wikipedia.org/wiki/Group_coded_recording "Group coded recording") tapes not only detects but also corrects single-bit errors. Some [file formats](https://en.wikipedia.org/wiki/File_format "File format"), particularly [archive formats](https://en.wikipedia.org/wiki/Archive_formats "Archive formats"), include a checksum (most often CRC-32\]) to detect corruption and truncation and can employ redundancy or [parity files](https://en.wikipedia.org/wiki/Parity_file "Parity file") to recover portions of corrupted data. [Reed-Solomon codes](https://en.wikipedia.org/wiki/Reed-Solomon_code "Reed-Solomon code") are used in [compact discs](https://en.wikipedia.org/wiki/Compact_disc "Compact disc") to correct errors caused by scratches. Modern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors.[\[21\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-21) [RAID](https://en.wikipedia.org/wiki/RAID "RAID") systems use a variety of error correction techniques to recover data when a hard drive completely fails. Filesystems such as [ZFS](https://en.wikipedia.org/wiki/ZFS "ZFS") or [Btrfs](https://en.wikipedia.org/wiki/Btrfs "Btrfs"), as well as some [RAID](https://en.wikipedia.org/wiki/RAID "RAID") implementations, support [data scrubbing](https://en.wikipedia.org/wiki/Data_scrubbing "Data scrubbing") and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used.[\[22\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-22) The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware. ### Error-correcting memory \[[edit](https://en.wikipedia.org/w/index.php?title=Error_detection_and_correction&action=edit&section=22 "Edit section: Error-correcting memory")\] [Dynamic random-access memory](https://en.wikipedia.org/wiki/Dynamic_random-access_memory "Dynamic random-access memory") (DRAM) may provide stronger protection against [soft errors](https://en.wikipedia.org/wiki/Soft_error "Soft error") by relying on error-correcting codes. Such error-correcting memory, known as ECC or EDAC-protected memory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased [radiation](https://en.wikipedia.org/wiki/Cosmic_ray "Cosmic ray") in space. Error-correcting memory controllers traditionally use [Hamming codes](https://en.wikipedia.org/wiki/Hamming_code "Hamming code"), although some use [triple modular redundancy](https://en.wikipedia.org/wiki/Triple_modular_redundancy "Triple modular redundancy"). [Interleaving](https://en.wikipedia.org/wiki/Error_correction_code#Interleaving "Error correction code") allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a [single-event upset](https://en.wikipedia.org/wiki/Single-event_upset "Single-event upset") (SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained.[\[23\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-23) In addition to hardware providing features required for ECC memory to operate, [operating systems](https://en.wikipedia.org/wiki/Operating_system "Operating system") usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the [Linux kernel](https://en.wikipedia.org/wiki/Linux_kernel "Linux kernel")'s EDAC subsystem (previously known as Bluesmoke), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the [PCI bus](https://en.wikipedia.org/wiki/PCI_bus "PCI bus").[\[24\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-24)[\[25\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-25)[\[26\]](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_note-26) A few systems\[[specify](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources "Wikipedia:Citing sources")\] also support [memory scrubbing](https://en.wikipedia.org/wiki/Memory_scrubbing "Memory scrubbing") to catch and correct errors early before they become unrecoverable. - [Berger code](https://en.wikipedia.org/wiki/Berger_code "Berger code") - [Burst error-correcting code](https://en.wikipedia.org/wiki/Burst_error-correcting_code "Burst error-correcting code") - [ECC memory](https://en.wikipedia.org/wiki/ECC_memory "ECC memory"), a type of computer data storage - [Link adaptation](https://en.wikipedia.org/wiki/Link_adaptation "Link adaptation") - [List of algorithms § Error detection and correction](https://en.wikipedia.org/wiki/List_of_algorithms#Error_detection_and_correction "List of algorithms") - [List of hash functions](https://en.wikipedia.org/wiki/List_of_hash_functions "List of hash functions") 1. ^ [*a](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-Jewish_1-0) [b](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-Jewish_1-1) "Masorah". [Jewish Encyclopedia](http://www.jewishencyclopedia.com/articles/10465-masorah). 2. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-2)* Pratico, Gary D.; Pelt, Miles V. Van (2009). [Basics of Biblical Hebrew Grammar: Second Edition](https://books.google.com/books?id=p3FIoT0s3yYC&pg=PT448). Zondervan. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-310-55882-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-310-55882-8 "Special:BookSources/978-0-310-55882-8") . 3. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-3) Mounce, William D. (2007). [Greek for the Rest of Us: Using Greek Tools Without Mastering Biblical Languages](https://books.google.com/books?id=AF-5ptJ0l2gC&pg=PA289). Zondervan. p. 289. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-310-28289-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-310-28289-1 "Special:BookSources/978-0-310-28289-1") . 4. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-4) [Mishneh Torah](https://en.wikipedia.org/wiki/Mishneh_Torah "Mishneh Torah"), Tefillin, Mezuzah, and Sefer Torah, 1:2. Example English translation: [Eliyahu Touger](https://en.wikipedia.org/w/index.php?title=Eliyahu_Touger&action=edit&redlink=1 "Eliyahu Touger (page does not exist)"). [The Rambam's Mishneh Torah](https://www.chabad.org/library/article_cdo/aid/925417/jewish/Tefillin-Mezuzah-and-Sefer-Torah-Chapter-One.htm). [Moznaim Publishing Corporation](https://en.wikipedia.org/w/index.php?title=Moznaim_Publishing_Corporation&action=edit&redlink=1 "Moznaim Publishing Corporation (page does not exist)"). 5. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-5) Brian M. Fagan (5 December 1996). "Dead Sea Scrolls". [The Oxford Companion to Archaeology](https://en.wikipedia.org/w/index.php?title=The_Oxford_Companion_to_Archaeology&action=edit&redlink=1 "The Oxford Companion to Archaeology (page does not exist)"). [Oxford University Press](https://en.wikipedia.org/wiki/Oxford_University_Press "Oxford University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0195076184](https://en.wikipedia.org/wiki/Special:BookSources/0195076184 "Special:BookSources/0195076184") . 6. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-Thompson_6-0) Thompson, Thomas M. (1983), From Error-Correcting Codes through Sphere Packings to Simple Groups, The Carus Mathematical Monographs (\#21), The Mathematical Association of America, p. vii, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-88385-023-0](https://en.wikipedia.org/wiki/Special:BookSources/0-88385-023-0 "Special:BookSources/0-88385-023-0") 7. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-7) Shannon, C.E. (1948), "A Mathematical Theory of Communication", Bell System Technical Journal, 27 (3): 379–423, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/j.1538-7305.1948.tb01338.x](https://doi.org/10.1002%2Fj.1538-7305.1948.tb01338.x), [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10338\.dmlcz/101429](https://hdl.handle.net/10338.dmlcz%2F101429), [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [9230594](https://pubmed.ncbi.nlm.nih.gov/9230594) 8. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-8) Golay, Marcel J. E. (1949), "Notes on Digital Coding", Proc.I.R.E. (I.E.E.E.), 37: 657 9. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-9) Gupta, Vikas; Verma, Chanderkant (November 2012). "Error Detection and Correction: An Introduction". International Journal of Advanced Research in Computer Science and Software Engineering. 2 (11). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [17499858](https://api.semanticscholar.org/CorpusID:17499858). 10. ^ [*a](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-reliable-erasure-code_10-0) [b](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-reliable-erasure-code_10-1) A. J. McAuley, Reliable Broadband Communication Using a Burst Erasure Correcting Code, ACM SIGCOMM, 1990. 11. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-11)* Shah, Pradeep M.; Vyavahare, Prakash D.; Jain, Anjana (September 2015). ["Modern error correcting codes for 4G and beyond: Turbo codes and LDPC codes"](https://www.researchgate.net/publication/301611980). 2015 Radio and Antenna Days of the Indian Ocean (RADIO). pp. 1–2\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/RADIO.2015.7323369](https://doi.org/10.1109%2FRADIO.2015.7323369). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-9-9903-7339-4](https://en.wikipedia.org/wiki/Special:BookSources/978-9-9903-7339-4 "Special:BookSources/978-9-9903-7339-4") . [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [28885076](https://api.semanticscholar.org/CorpusID:28885076). Retrieved 22 May 2022. 12. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-12) ["IEEE SA - IEEE 802.11ac-2013"](https://web.archive.org/web/20220522205452/https://standards.ieee.org/ieee/802.11ac/4473/). IEEE Standards Association. Archived from [the original](https://standards.ieee.org/ieee/802.11ac/4473/) on 2022-05-22. Retrieved 2022-05-22. 13. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-13) ["Transition to Advanced Format 4K Sector Hard Drives \\| Seagate US"](https://www.seagate.com/sg/en/tech-insights/advanced-format-4k-sector-hard-drives-master-ti/). Seagate.com. Retrieved 22 May 2022. 14. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-14) Frank van Gerwen. ["Numbers (and other mysterious) stations"](https://web.archive.org/web/20170712053335/http://www.cisquet.nl/numbers.htm). Archived from [the original](http://www.cisquet.nl/numbers.htm) on 12 July 2017. Retrieved 12 March 2012. 15. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-15) Gary Cutlack (25 August 2010). ["Mysterious Russian 'Numbers Station' Changes Broadcast After 20 Years"](https://web.archive.org/web/20170705131816/https://www.gizmodo.com.au/2010/08/mysterious-russian-numbers-station-changes-broadcast-after-20-years/). Gizmodo. Archived from [the original](http://www.gizmodo.com.au/2010/08/mysterious-russian-numbers-station-changes-broadcast-after-20-years/) on 5 July 2017. Retrieved 12 March 2012. 16. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-16) Ben-Gal I.; Herer Y.; Raz T. (2003). ["Self-correcting inspection procedure under inspection errors"](https://web.archive.org/web/20131013171945/http://www.eng.tau.ac.il/~bengal/SCI_paper.pdf) (PDF). IIE Transactions. IIE Transactions on Quality and Reliability, 34(6), pp. 529-540. Archived from [the original](http://www.eng.tau.ac.il/~bengal/SCI_paper.pdf) (PDF) on 2013-10-13. Retrieved 2014-01-10. 17. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-DOI10.7275/bjvb-2n37_17-0) Yvo Meeres, Tommi A. Pirinen (2021). "Vowel Harmony Viewed as Error-Correcting Code". Proceedings of the Society for Computation in Linguistics. 4 (1): 313–322\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.7275/bjvb-2n37](https://doi.org/10.7275%2Fbjvb-2n37). 18. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-deep-space-codes_18-0) K. Andrews et al., The Development of Turbo and LDPC Codes for Deep-Space Applications, Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007. 19. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-voyager_19-0) Huffman, William Cary; [Pless, Vera S.](https://en.wikipedia.org/wiki/Vera_Pless "Vera Pless") (2003). [Fundamentals of Error-Correcting Codes](https://archive.org/details/fundamentalsofer0000huff). [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-521-78280-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-78280-7 "Special:BookSources/978-0-521-78280-7") . 20. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-20) Kurtas, Erozan M.; Vasic, Bane (2018-10-03). Advanced Error Control Techniques for Data Storage Systems. CRC Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4200-3649-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4200-3649-7 "Special:BookSources/978-1-4200-3649-7") . 21. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-21) Scott A. Moulton. ["My Hard Drive Died"](https://web.archive.org/web/20080202143103/http://www.myharddrivedied.com/presentations_whitepaper.html). Archived from [the original](http://www.myharddrivedied.com/presentations_whitepaper.html) on 2008-02-02. 22. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-22) Qiao, Zhi; Fu, Song; Chen, Hsing-Bung; Settlemyer, Bradley (2019). "Building Reliable High-Performance Storage Systems: An Empirical and Analytical Study". 2019 IEEE International Conference on Cluster Computing (CLUSTER). pp. 1–10\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/CLUSTER.2019.8891006](https://doi.org/10.1109%2FCLUSTER.2019.8891006). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-7281-4734-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-7281-4734-5 "Special:BookSources/978-1-7281-4734-5") . [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [207951690](https://api.semanticscholar.org/CorpusID:207951690). 23. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-23) ["Using StrongArm SA-1110 in the On-Board Computer of Nanosatellite"](https://web.archive.org/web/20111002152735/http://www.apmcsta.org/File/doc/Conferences/6th%20meeting/Chen%20Zhenyu.doc). Tsinghua Space Center, [Tsinghua University](https://en.wikipedia.org/wiki/Tsinghua_University "Tsinghua University"), Beijing. Archived from [the original](http://www.apmcsta.org/File/doc/Conferences/6th%20meeting/Chen%20Zhenyu.doc) on 2011-10-02. Retrieved 2009-02-16. 24. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-24) Jeff Layton. ["Error Detection and Correction"](http://www.admin-magazine.com/Articles/Monitoring-Memory-Errors). [Linux Magazine](https://en.wikipedia.org/wiki/Linux_Magazine "Linux Magazine"). Retrieved 2014-08-12. 25. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-25) ["EDAC Project"](https://bluesmoke.sourceforge.net/). bluesmoke.sourceforge.net. Retrieved 2014-08-12. 26. [^](https://en.wikipedia.org/wiki/Error_detection_and_correction#cite_ref-26) ["Documentation/edac.txt"](https://web.archive.org/web/20090905174616/http://www.kernel.org/doc/Documentation/edac.txt). Linux kernel documentation. [kernel.org](https://en.wikipedia.org/wiki/Kernel.org "Kernel.org"). 2014-06-16. Archived from [the original](https://www.kernel.org/doc/Documentation/edac.txt) on 2009-09-05. Retrieved 2014-08-12. - Shu Lin; Daniel J. Costello, Jr. (1983). Error Control Coding: Fundamentals and Applications. [Prentice Hall](https://en.wikipedia.org/wiki/Prentice_Hall "Prentice Hall"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-13-283796-X](https://en.wikipedia.org/wiki/Special:BookSources/0-13-283796-X "Special:BookSources/0-13-283796-X") . - [SoftECC: A System for Software Memory Integrity Checking](http://pdos.csail.mit.edu/papers/softecc:ddopson-meng/softecc_ddopson-meng.pdf) - [A Tunable, Software-based DRAM Error Detection and Correction Library for HPC](http://www.fiala.me/pubs/papers/libsdc11.pdf) [Archived](https://web.archive.org/web/20141107074502/http://www.fiala.me/pubs/papers/libsdc11.pdf) 2014-11-07 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") - [Detection and Correction of Silent Data Corruption for Large-Scale High-Performance Computing](http://www.fiala.me/pubs/papers/sc12-redmpi.pdf) [Archived](https://web.archive.org/web/20141107074511/http://www.fiala.me/pubs/papers/sc12-redmpi.pdf) 2014-11-07 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") - [The on-line textbook: Information Theory, Inference, and Learning Algorithms](http://www.inference.phy.cam.ac.uk/mackay/itila/), by [David J.C. MacKay](https://en.wikipedia.org/wiki/David_J.C._MacKay "David J.C. MacKay"), contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including [low-density parity-check codes](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code"), [turbo codes](https://en.wikipedia.org/wiki/Turbo_code "Turbo code"), and [fountain codes](https://en.wikipedia.org/wiki/Fountain_codes "Fountain codes"). - [ECC Page](http://www.eccpage.com/) – implementations of popular ECC encoding and decoding routines
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