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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 184 (from laksa098)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa184.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://mathworld.wolfram.com/Error-CorrectingCode.html\')), getAhrefsUnparsedNoserviceFromURL(\'https://mathworld.wolfram.com/Error-CorrectingCode.html\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/mathworld.wolfram.com\/Error-CorrectingCode.html","crawl_time":1776013008,"first_indexed_time":1587268868,"http_code":200,"src_unparsed":"com,wolfram!mathworld,\/Error-CorrectingCode.html s443","src_root_hash":"3744487911316863784","history_drop_reason":null,"meta_title":"Error-Correcting Code -- from Wolfram MathWorld","meta_descriptions":["An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as coding theory. Error detection is much simpler than error correction, and one or more \"check\" digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes..."],"attrs_boilerpipe_text":"Algebra\nApplied Mathematics\nCalculus and Analysis\nDiscrete Mathematics\nFoundations of Mathematics\nGeometry\nHistory and Terminology\nNumber Theory\nProbability and Statistics\nRecreational Mathematics\nTopology\nAlphabetical Index\nNew in MathWorld\nAn error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain\n limitations) based on the remaining numbers. The study of error-correcting codes\n and the associated mathematics is known as\ncoding theory\n.\nError detection is much simpler than error correction, and one or more \"check\" digits are commonly embedded in credit card numbers in order to detect mistakes.\n Early space probes like Mariner used a type of error-correcting code called a block\n code, and more recent space probes use convolution codes. Error-correcting codes\n are also used in CD players, high speed modems, and cellular phones. Modems use error\n detection when they compute\nchecksums\n, which are sums\n of the digits in a given transmission modulo some number. The\nISBN\nused to identify books also incorporates a check\ndigit\n.\nA powerful check for 13\ndigit\nnumbers consists of the following. Write the number as a string of\ndigits\n. Take\nand double. Now add the number of\ndigits\nin\nodd\npositions which are\nto this number. Now add\n. The check number is then the number required\n to bring the last\ndigit\nto 0. This scheme detects all single\ndigit\nerrors and all\ntranspositions\nof adjacent\ndigits\nexcept 0 and 9.\nLet\ndenote the maximal number of\n(0,1)-vectors having the property that any two of the set\n differ in at least\nplaces. The corresponding vectors can correct\nerrors.\nis the number of\ns with precisely\n1s (Sloane and Plouffe 1995). Since it is not possible for\n-vectors\n to differ in\nplaces and since\n-vectors\n which differ in all\nplaces partition into disparate sets of two,\n(1)\nValues of\ncan be found by labeling the\n(0,1)-\n-vectors, finding all unordered pairs\nof\n-vectors which differ from each other in at least\nplaces, forming a\ngraph\nfrom these\n unordered pairs, and then finding the\nclique number\nof this graph. Unfortunately, finding the size of a clique for a given\ngraph\nis an\nNP-complete problem\n.\nOEIS\n1\nA000079\n2, 4, 8, 16, 32,\n 64, 128, ...\n2\n1, 2, 4, 8, ...\n3\n1, 1, 2, 2, ...\n4\nA005864\n1, 1, 1, 2, 4, 8,\n 16, 20, 40, ...\n5\n1, 1, 1, 1, 2, ...\n6\nA005865\n1,\n 1, 1, 1, 1, 2, 2, 2, 4, 6, 12, ...\n7\n1, 1, 1, 1, 1, 1, 2, ...\n8\nA005866\n1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, ...\nSee also\nChecksum\n,\nClique\n,\nClique Number\n,\nCoding\n Theory\n,\nFinite Field\n,\nGolay\n Code\n,\nHadamard Matrix\n,\nHalved\n Cube Graph\n,\nHamming Code\n,\nISBN\n,\nPerfect Code\n,\nUPC\nExplore with Wolfram|Alpha\nReferences\nBaylis, J.\nError Correcting Codes: A Mathematical Introduction.\nBoca Raton, FL: CRC Press,\n 1998.\nBerlekamp, E. R.\nAlgebraic\n Coding Theory, rev. ed.\nNew York: McGraw-Hill, 1968.\nBrouwer,\n A. E.; Shearer, J. B.; Sloane, N. J. A.; and Smith, W. D.\n \"A New Table of Constant Weight Codes.\"\nIEEE Trans. Inform. Th.\n36\n,\n 1334-1380, 1990.\nCalderbank, A. R.; Hammons, A. R. Jr.; Kumar,\n P. V.; Sloane, N. J. A.; and Solé, P. \"A Linear Construction\n for Certain Kerdock and Preparata Codes.\"\nBull. Amer. Math. Soc.\n29\n,\n 218-222, 1993.\nConway, J. H. and Sloane, N. J. A. \"Quaternary\n Constructions for the Binary Single-Error-Correcting Codes of Julin, Best and Others.\"\nDes. Codes Cryptogr.\n4\n, 31-42, 1994.\nConway, J. H.\n and Sloane, N. J. A. \"Error-Correcting Codes.\" §3.2 in\nSphere\n Packings, Lattices, and Groups, 2nd ed.\nNew York: Springer-Verlag, pp. 75-88,\n 1993.\nGachkov, I. \"Error-Correcting Codes with\nMathematica\n.\"\nhttp:\/\/library.wolfram.com\/infocenter\/MathSource\/5085\/\n.\nGallian,\n J. \"How Computers Can Read and Correct ID Numbers.\"\nMath Horizons\n,\n pp. 14-15, Winter 1993.\nGuy, R. K.\nUnsolved\n Problems in Number Theory, 2nd ed.\nNew York: Springer-Verlag, pp. 119-121,\n 1994.\nMacWilliams, F. J. and Sloane, N. J. A.\nThe\n Theory of Error-Correcting Codes.\nAmsterdam, Netherlands: North-Holland,\n 1977.\nSloane, N. J. A. Sequences\nA000079\n\/M1129,\nA005864\n\/M1111,\nA005865\n\/M0240,\n and\nA005866\n\/M0226 in \"The On-Line Encyclopedia\n of Integer Sequences.\"\nSloane, N. J. A. and Plouffe, S.\n Figure M0240 in\nThe\n Encyclopedia of Integer Sequences.\nSan Diego: Academic Press, 1995.\nReferenced\n on Wolfram|Alpha\nError-Correcting Code\nCite this as:\nWeisstein, Eric W.\n\"Error-Correcting Code.\"\nFrom\nMathWorld\n--A Wolfram Resource.\nhttps:\/\/mathworld.wolfram.com\/Error-CorrectingCode.html\nSubject classifications","attrs_markdown":"![](https:\/\/mathworld.wolfram.com\/images\/header\/menu-icon.png) TOPICS\n\n![](https:\/\/mathworld.wolfram.com\/images\/sidebar\/menu-close.png)\n\n[Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)   [Applied Mathematics](https:\/\/mathworld.wolfram.com\/topics\/AppliedMathematics.html)   [Calculus and Analysis](https:\/\/mathworld.wolfram.com\/topics\/CalculusandAnalysis.html)   [Discrete Mathematics](https:\/\/mathworld.wolfram.com\/topics\/DiscreteMathematics.html)   [Foundations of Mathematics](https:\/\/mathworld.wolfram.com\/topics\/FoundationsofMathematics.html)   [Geometry](https:\/\/mathworld.wolfram.com\/topics\/Geometry.html)   [History and Terminology](https:\/\/mathworld.wolfram.com\/topics\/HistoryandTerminology.html)   [Number Theory](https:\/\/mathworld.wolfram.com\/topics\/NumberTheory.html)   [Probability and Statistics](https:\/\/mathworld.wolfram.com\/topics\/ProbabilityandStatistics.html)   [Recreational Mathematics](https:\/\/mathworld.wolfram.com\/topics\/RecreationalMathematics.html)   [Topology](https:\/\/mathworld.wolfram.com\/topics\/Topology.html)\n\n[Alphabetical Index](https:\/\/mathworld.wolfram.com\/letters\/)   [New in MathWorld](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n\n- [Discrete Mathematics](https:\/\/mathworld.wolfram.com\/topics\/DiscreteMathematics.html)\n- [Coding Theory](https:\/\/mathworld.wolfram.com\/topics\/CodingTheory.html)\n# Error-Correcting Code\n***\nAn error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as [coding theory](https:\/\/mathworld.wolfram.com\/CodingTheory.html).\n\nError detection is much simpler than error correction, and one or more \"check\" digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes like Mariner used a type of error-correcting code called a block code, and more recent space probes use convolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modems use error detection when they compute [checksums](https:\/\/mathworld.wolfram.com\/Checksum.html), which are sums of the digits in a given transmission modulo some number. The [ISBN](https:\/\/mathworld.wolfram.com\/ISBN.html) used to identify books also incorporates a check [digit](https:\/\/mathworld.wolfram.com\/Digit.html).\n\nA powerful check for 13 [digit](https:\/\/mathworld.wolfram.com\/Digit.html) numbers consists of the following. Write the number as a string of [digits](https:\/\/mathworld.wolfram.com\/Digit.html) ![a\\_1a\\_2a\\_3...a\\_(13)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline1.svg). Take ![a\\_1+a\\_3+...+a\\_(13)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline2.svg) and double. Now add the number of [digits](https:\/\/mathworld.wolfram.com\/Digit.html) in [odd](https:\/\/mathworld.wolfram.com\/OddNumber.html) positions which are ![\\>4](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline3.svg) to this number. Now add ![a\\_2+a\\_4+...+a\\_(12)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline4.svg). The check number is then the number required to bring the last [digit](https:\/\/mathworld.wolfram.com\/Digit.html) to 0. This scheme detects all single [digit](https:\/\/mathworld.wolfram.com\/Digit.html) errors and all [transpositions](https:\/\/mathworld.wolfram.com\/Transposition.html) of adjacent [digits](https:\/\/mathworld.wolfram.com\/Digit.html) except 0 and 9.\n\nLet ![A(n,d)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline5.svg) denote the maximal number of ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline6.svg) (0,1)-vectors having the property that any two of the set differ in at least ![d](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline7.svg) places. The corresponding vectors can correct ![\\[(d-1)\/2\\]](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline8.svg) errors. ![A(n,d,w)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline9.svg) is the number of ![A(n,d)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline10.svg)s with precisely ![w](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline11.svg) 1s (Sloane and Plouffe 1995). Since it is not possible for ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline12.svg)\\-vectors to differ in ![d\\>n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline13.svg) places and since ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline14.svg)\\-vectors which differ in all ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline15.svg) places partition into disparate sets of two,\n\n|   |   |\n|---|---|\n| ![ A(n,d)={1 n\\<d; 2 n=d. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/NumberedEquation1.svg) | (1) |\n\nValues of ![A(n,d)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline16.svg) can be found by labeling the ![2^n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline17.svg) (0,1)-![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline18.svg)\\-vectors, finding all unordered pairs ![(a\\_i,a\\_j)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline19.svg) of ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline20.svg)\\-vectors which differ from each other in at least ![d](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline21.svg) places, forming a [graph](https:\/\/mathworld.wolfram.com\/Graph.html) from these unordered pairs, and then finding the [clique number](https:\/\/mathworld.wolfram.com\/CliqueNumber.html) of this graph. Unfortunately, finding the size of a clique for a given [graph](https:\/\/mathworld.wolfram.com\/Graph.html) is an [NP-complete problem](https:\/\/mathworld.wolfram.com\/NP-CompleteProblem.html).\n\n|   |   |   |\n|---|---|---|\n| ![d](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline22.svg) | OEIS | ![A(n,d)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline23.svg) |\n| 1 | [A000079](http:\/\/oeis.org\/A000079) | 2, 4, 8, 16, 32, 64, 128, ... |\n| 2 |   | 1, 2, 4, 8, ... |\n| 3 |   | 1, 1, 2, 2, ... |\n| 4 | [A005864](http:\/\/oeis.org\/A005864) | 1, 1, 1, 2, 4, 8, 16, 20, 40, ... |\n| 5 |   | 1, 1, 1, 1, 2, ... |\n| 6 | [A005865](http:\/\/oeis.org\/A005865) | 1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 12, ... |\n| 7 |   | 1, 1, 1, 1, 1, 1, 2, ... |\n| 8 | [A005866](http:\/\/oeis.org\/A005866) | 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, ... |\n***\n## See also\n[Checksum](https:\/\/mathworld.wolfram.com\/Checksum.html), [Clique](https:\/\/mathworld.wolfram.com\/Clique.html), [Clique Number](https:\/\/mathworld.wolfram.com\/CliqueNumber.html), [Coding Theory](https:\/\/mathworld.wolfram.com\/CodingTheory.html), [Finite Field](https:\/\/mathworld.wolfram.com\/FiniteField.html), [Golay Code](https:\/\/mathworld.wolfram.com\/GolayCode.html), [Hadamard Matrix](https:\/\/mathworld.wolfram.com\/HadamardMatrix.html), [Halved Cube Graph](https:\/\/mathworld.wolfram.com\/HalvedCubeGraph.html), [Hamming Code](https:\/\/mathworld.wolfram.com\/HammingCode.html), [ISBN](https:\/\/mathworld.wolfram.com\/ISBN.html), [Perfect Code](https:\/\/mathworld.wolfram.com\/PerfectCode.html), [UPC](https:\/\/mathworld.wolfram.com\/UPC.html)\n\n## Explore with Wolfram\\|Alpha\n![WolframAlpha](https:\/\/mathworld.wolfram.com\/images\/wolframalpha\/WA-logo.png)\n\nMore things to try:\n\n- [BCH code](https:\/\/www.wolframalpha.com\/input\/?i=BCH+code)\n- [coding theory](https:\/\/www.wolframalpha.com\/input\/?i=coding+theory)\n- [.999... = 1](https:\/\/www.wolframalpha.com\/input\/?i=.999...+%3D+1)\n## References\nBaylis, J. *[Error Correcting Codes: A Mathematical Introduction.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0412786907\/ref=nosim\/ericstreasuretro)* Boca Raton, FL: CRC Press, 1998.\n\nBerlekamp, E. R. *[Algebraic Coding Theory, rev. ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0894120638\/ref=nosim\/ericstreasuretro)* New York: McGraw-Hill, 1968.\n\nBrouwer, A. E.; Shearer, J. B.; Sloane, N. J. A.; and Smith, W. D. \"A New Table of Constant Weight Codes.\" *IEEE Trans. Inform. Th.* **36**, 1334-1380, 1990.\n\nCalderbank, A. R.; Hammons, A. R. Jr.; Kumar, P. V.; Sloane, N. J. A.; and Solé, P. \"A Linear Construction for Certain Kerdock and Preparata Codes.\" *Bull. Amer. Math. Soc.* **29**, 218-222, 1993.\n\nConway, J. H. and Sloane, N. J. A. \"Quaternary Constructions for the Binary Single-Error-Correcting Codes of Julin, Best and Others.\" *Des. Codes Cryptogr.* **4**, 31-42, 1994.\n\nConway, J. H. and Sloane, N. J. A. \"Error-Correcting Codes.\" §3.2 in *[Sphere Packings, Lattices, and Groups, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0387979123\/ref=nosim\/ericstreasuretro)* New York: Springer-Verlag, pp. 75-88, 1993.\n\n![](https:\/\/mathworld.wolfram.com\/images\/entries\/mathematica.gif) Gachkov, I. \"Error-Correcting Codes with *Mathematica*.\" <http:\/\/library.wolfram.com\/infocenter\/MathSource\/5085\/>.\n\nGallian, J. \"How Computers Can Read and Correct ID Numbers.\" *Math Horizons*, pp. 14-15, Winter 1993.\n\nGuy, R. K. *[Unsolved Problems in Number Theory, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0387208607\/ref=nosim\/ericstreasuretro)* New York: Springer-Verlag, pp. 119-121, 1994.\n\nMacWilliams, F. J. and Sloane, N. J. A. *[The Theory of Error-Correcting Codes.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0444851933\/ref=nosim\/ericstreasuretro)* Amsterdam, Netherlands: North-Holland, 1977.\n\nSloane, N. J. A. Sequences [A000079](http:\/\/oeis.org\/A000079)\/M1129, [A005864](http:\/\/oeis.org\/A005864)\/M1111, [A005865](http:\/\/oeis.org\/A005865)\/M0240, and [A005866](http:\/\/oeis.org\/A005866)\/M0226 in \"The On-Line Encyclopedia of Integer Sequences.\"\n\nSloane, N. J. A. and Plouffe, S. Figure M0240 in *[The Encyclopedia of Integer Sequences.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0125586302\/ref=nosim\/ericstreasuretro)* San Diego: Academic Press, 1995.\n\n## Referenced on Wolfram\\|Alpha\n[Error-Correcting Code](https:\/\/www.wolframalpha.com\/input\/?i=error-correcting+code \"Error-Correcting Code\")\n\n## Cite this as:\n[Weisstein, Eric W.](https:\/\/mathworld.wolfram.com\/about\/author.html) \"Error-Correcting Code.\" From [*MathWorld*](https:\/\/mathworld.wolfram.com\/)\\--A Wolfram Resource. <https:\/\/mathworld.wolfram.com\/Error-CorrectingCode.html>\n\n## Subject classifications\n- [Discrete Mathematics](https:\/\/mathworld.wolfram.com\/topics\/DiscreteMathematics.html)\n- [Coding Theory](https:\/\/mathworld.wolfram.com\/topics\/CodingTheory.html)\n\n- [About MathWorld](https:\/\/mathworld.wolfram.com\/about\/)\n- [MathWorld Classroom](https:\/\/mathworld.wolfram.com\/classroom\/)\n- [Contribute](https:\/\/mathworld.wolfram.com\/contact\/)\n- [MathWorld Book](https:\/\/www.amazon.com\/exec\/obidos\/ASIN\/1420072218\/ref=nosim\/weisstein-20)\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n\n- [13,311 Entries](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n- [Last Updated: Wed Mar 25 2026](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n- [©1999–2026 Wolfram Research, Inc.](https:\/\/www.wolfram.com\/)\n- [Terms of Use](https:\/\/www.wolfram.com\/legal\/terms\/mathworld.html)\n\n- [![Wolfram](https:\/\/mathworld.wolfram.com\/images\/footer\/wolfram-logo.png)](https:\/\/www.wolfram.com\/)\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n- [Wolfram for Education](https:\/\/www.wolfram.com\/education\/)\n- Created, developed and nurtured by Eric Weisstein at Wolfram Research\n\n*Created, developed and nurtured by Eric Weisstein at Wolfram Research*\n\n[Find out if you already have access to Wolfram tech through your organization](https:\/\/www.wolfram.com\/siteinfo\/)\n\n×","attrs_readable_markdown":"[Algebra](https:\/\/mathworld.wolfram.com\/topics\/Algebra.html)   [Applied Mathematics](https:\/\/mathworld.wolfram.com\/topics\/AppliedMathematics.html)   [Calculus and Analysis](https:\/\/mathworld.wolfram.com\/topics\/CalculusandAnalysis.html)   [Discrete Mathematics](https:\/\/mathworld.wolfram.com\/topics\/DiscreteMathematics.html)   [Foundations of Mathematics](https:\/\/mathworld.wolfram.com\/topics\/FoundationsofMathematics.html)   [Geometry](https:\/\/mathworld.wolfram.com\/topics\/Geometry.html)   [History and Terminology](https:\/\/mathworld.wolfram.com\/topics\/HistoryandTerminology.html)   [Number Theory](https:\/\/mathworld.wolfram.com\/topics\/NumberTheory.html)   [Probability and Statistics](https:\/\/mathworld.wolfram.com\/topics\/ProbabilityandStatistics.html)   [Recreational Mathematics](https:\/\/mathworld.wolfram.com\/topics\/RecreationalMathematics.html)   [Topology](https:\/\/mathworld.wolfram.com\/topics\/Topology.html)\n\n[Alphabetical Index](https:\/\/mathworld.wolfram.com\/letters\/)   [New in MathWorld](https:\/\/mathworld.wolfram.com\/whatsnew\/)\n\n***\nAn error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as [coding theory](https:\/\/mathworld.wolfram.com\/CodingTheory.html).\n\nError detection is much simpler than error correction, and one or more \"check\" digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes like Mariner used a type of error-correcting code called a block code, and more recent space probes use convolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modems use error detection when they compute [checksums](https:\/\/mathworld.wolfram.com\/Checksum.html), which are sums of the digits in a given transmission modulo some number. The [ISBN](https:\/\/mathworld.wolfram.com\/ISBN.html) used to identify books also incorporates a check [digit](https:\/\/mathworld.wolfram.com\/Digit.html).\n\nA powerful check for 13 [digit](https:\/\/mathworld.wolfram.com\/Digit.html) numbers consists of the following. Write the number as a string of [digits](https:\/\/mathworld.wolfram.com\/Digit.html) ![a\\_1a\\_2a\\_3...a\\_(13)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline1.svg). Take ![a\\_1+a\\_3+...+a\\_(13)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline2.svg) and double. Now add the number of [digits](https:\/\/mathworld.wolfram.com\/Digit.html) in [odd](https:\/\/mathworld.wolfram.com\/OddNumber.html) positions which are ![\\>4](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline3.svg) to this number. Now add ![a\\_2+a\\_4+...+a\\_(12)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline4.svg). The check number is then the number required to bring the last [digit](https:\/\/mathworld.wolfram.com\/Digit.html) to 0. This scheme detects all single [digit](https:\/\/mathworld.wolfram.com\/Digit.html) errors and all [transpositions](https:\/\/mathworld.wolfram.com\/Transposition.html) of adjacent [digits](https:\/\/mathworld.wolfram.com\/Digit.html) except 0 and 9.\n\nLet ![A(n,d)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline5.svg) denote the maximal number of ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline6.svg) (0,1)-vectors having the property that any two of the set differ in at least ![d](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline7.svg) places. The corresponding vectors can correct ![\\[(d-1)\/2\\]](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline8.svg) errors. ![A(n,d,w)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline9.svg) is the number of ![A(n,d)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline10.svg)s with precisely ![w](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline11.svg) 1s (Sloane and Plouffe 1995). Since it is not possible for ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline12.svg)\\-vectors to differ in ![d\\>n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline13.svg) places and since ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline14.svg)\\-vectors which differ in all ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline15.svg) places partition into disparate sets of two,\n\n|   |   |\n|---|---|\n| ![ A(n,d)={1 n\\<d; 2 n=d. ](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/NumberedEquation1.svg) | (1) |\n\nValues of ![A(n,d)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline16.svg) can be found by labeling the ![2^n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline17.svg) (0,1)-![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline18.svg)\\-vectors, finding all unordered pairs ![(a\\_i,a\\_j)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline19.svg) of ![n](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline20.svg)\\-vectors which differ from each other in at least ![d](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline21.svg) places, forming a [graph](https:\/\/mathworld.wolfram.com\/Graph.html) from these unordered pairs, and then finding the [clique number](https:\/\/mathworld.wolfram.com\/CliqueNumber.html) of this graph. Unfortunately, finding the size of a clique for a given [graph](https:\/\/mathworld.wolfram.com\/Graph.html) is an [NP-complete problem](https:\/\/mathworld.wolfram.com\/NP-CompleteProblem.html).\n\n|   |   |   |\n|---|---|---|\n| ![d](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline22.svg) | OEIS | ![A(n,d)](https:\/\/mathworld.wolfram.com\/images\/equations\/Error-CorrectingCode\/Inline23.svg) |\n| 1 | [A000079](http:\/\/oeis.org\/A000079) | 2, 4, 8, 16, 32, 64, 128, ... |\n| 2 |   | 1, 2, 4, 8, ... |\n| 3 |   | 1, 1, 2, 2, ... |\n| 4 | [A005864](http:\/\/oeis.org\/A005864) | 1, 1, 1, 2, 4, 8, 16, 20, 40, ... |\n| 5 |   | 1, 1, 1, 1, 2, ... |\n| 6 | [A005865](http:\/\/oeis.org\/A005865) | 1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 12, ... |\n| 7 |   | 1, 1, 1, 1, 1, 1, 2, ... |\n| 8 | [A005866](http:\/\/oeis.org\/A005866) | 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, ... |\n***\n## See also\n[Checksum](https:\/\/mathworld.wolfram.com\/Checksum.html), [Clique](https:\/\/mathworld.wolfram.com\/Clique.html), [Clique Number](https:\/\/mathworld.wolfram.com\/CliqueNumber.html), [Coding Theory](https:\/\/mathworld.wolfram.com\/CodingTheory.html), [Finite Field](https:\/\/mathworld.wolfram.com\/FiniteField.html), [Golay Code](https:\/\/mathworld.wolfram.com\/GolayCode.html), [Hadamard Matrix](https:\/\/mathworld.wolfram.com\/HadamardMatrix.html), [Halved Cube Graph](https:\/\/mathworld.wolfram.com\/HalvedCubeGraph.html), [Hamming Code](https:\/\/mathworld.wolfram.com\/HammingCode.html), [ISBN](https:\/\/mathworld.wolfram.com\/ISBN.html), [Perfect Code](https:\/\/mathworld.wolfram.com\/PerfectCode.html), [UPC](https:\/\/mathworld.wolfram.com\/UPC.html)\n\n## Explore with Wolfram\\|Alpha\n## References\nBaylis, J. *[Error Correcting Codes: A Mathematical Introduction.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0412786907\/ref=nosim\/ericstreasuretro)* Boca Raton, FL: CRC Press, 1998.Berlekamp, E. R. *[Algebraic Coding Theory, rev. ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0894120638\/ref=nosim\/ericstreasuretro)* New York: McGraw-Hill, 1968.Brouwer, A. E.; Shearer, J. B.; Sloane, N. J. A.; and Smith, W. D. \"A New Table of Constant Weight Codes.\" *IEEE Trans. Inform. Th.* **36**, 1334-1380, 1990.Calderbank, A. R.; Hammons, A. R. Jr.; Kumar, P. V.; Sloane, N. J. A.; and Solé, P. \"A Linear Construction for Certain Kerdock and Preparata Codes.\" *Bull. Amer. Math. Soc.* **29**, 218-222, 1993.Conway, J. H. and Sloane, N. J. A. \"Quaternary Constructions for the Binary Single-Error-Correcting Codes of Julin, Best and Others.\" *Des. Codes Cryptogr.* **4**, 31-42, 1994.Conway, J. H. and Sloane, N. J. A. \"Error-Correcting Codes.\" §3.2 in *[Sphere Packings, Lattices, and Groups, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0387979123\/ref=nosim\/ericstreasuretro)* New York: Springer-Verlag, pp. 75-88, 1993.![](https:\/\/mathworld.wolfram.com\/images\/entries\/mathematica.gif) Gachkov, I. \"Error-Correcting Codes with *Mathematica*.\" <http:\/\/library.wolfram.com\/infocenter\/MathSource\/5085\/>.Gallian, J. \"How Computers Can Read and Correct ID Numbers.\" *Math Horizons*, pp. 14-15, Winter 1993.Guy, R. K. *[Unsolved Problems in Number Theory, 2nd ed.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0387208607\/ref=nosim\/ericstreasuretro)* New York: Springer-Verlag, pp. 119-121, 1994.MacWilliams, F. J. and Sloane, N. J. A. *[The Theory of Error-Correcting Codes.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0444851933\/ref=nosim\/ericstreasuretro)* Amsterdam, Netherlands: North-Holland, 1977.Sloane, N. J. A. Sequences [A000079](http:\/\/oeis.org\/A000079)\/M1129, [A005864](http:\/\/oeis.org\/A005864)\/M1111, [A005865](http:\/\/oeis.org\/A005865)\/M0240, and [A005866](http:\/\/oeis.org\/A005866)\/M0226 in \"The On-Line Encyclopedia of Integer Sequences.\"Sloane, N. J. A. and Plouffe, S. Figure M0240 in *[The Encyclopedia of Integer Sequences.](http:\/\/www.amazon.com\/exec\/obidos\/ASIN\/0125586302\/ref=nosim\/ericstreasuretro)* San Diego: Academic Press, 1995.\n\n## Referenced on Wolfram\\|Alpha\n[Error-Correcting Code](https:\/\/www.wolframalpha.com\/input\/?i=error-correcting+code \"Error-Correcting Code\")\n\n## Cite this as:\n[Weisstein, Eric W.](https:\/\/mathworld.wolfram.com\/about\/author.html) \"Error-Correcting Code.\" From [*MathWorld*](https:\/\/mathworld.wolfram.com\/)\\--A Wolfram Resource. <https:\/\/mathworld.wolfram.com\/Error-CorrectingCode.html>\n\n## Subject classifications","meta_canonical":null}

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Meta Description	An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as coding theory. Error detection is much simpler than error correction, and one or more "check" digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes...
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Boilerpipe Text	Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as coding theory . Error detection is much simpler than error correction, and one or more "check" digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes like Mariner used a type of error-correcting code called a block code, and more recent space probes use convolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modems use error detection when they compute checksums , which are sums of the digits in a given transmission modulo some number. The ISBN used to identify books also incorporates a check digit . A powerful check for 13 digit numbers consists of the following. Write the number as a string of digits . Take and double. Now add the number of digits in odd positions which are to this number. Now add . The check number is then the number required to bring the last digit to 0. This scheme detects all single digit errors and all transpositions of adjacent digits except 0 and 9. Let denote the maximal number of (0,1)-vectors having the property that any two of the set differ in at least places. The corresponding vectors can correct errors. is the number of s with precisely 1s (Sloane and Plouffe 1995). Since it is not possible for -vectors to differ in places and since -vectors which differ in all places partition into disparate sets of two, (1) Values of can be found by labeling the (0,1)- -vectors, finding all unordered pairs of -vectors which differ from each other in at least places, forming a graph from these unordered pairs, and then finding the clique number of this graph. Unfortunately, finding the size of a clique for a given graph is an NP-complete problem . OEIS 1 A000079 2, 4, 8, 16, 32, 64, 128, ... 2 1, 2, 4, 8, ... 3 1, 1, 2, 2, ... 4 A005864 1, 1, 1, 2, 4, 8, 16, 20, 40, ... 5 1, 1, 1, 1, 2, ... 6 A005865 1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 12, ... 7 1, 1, 1, 1, 1, 1, 2, ... 8 A005866 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, ... See also Checksum , Clique , Clique Number , Coding Theory , Finite Field , Golay Code , Hadamard Matrix , Halved Cube Graph , Hamming Code , ISBN , Perfect Code , UPC Explore with Wolfram\|Alpha References Baylis, J. Error Correcting Codes: A Mathematical Introduction. Boca Raton, FL: CRC Press, 1998. Berlekamp, E. R. Algebraic Coding Theory, rev. ed. New York: McGraw-Hill, 1968. Brouwer, A. E.; Shearer, J. B.; Sloane, N. J. A.; and Smith, W. D. "A New Table of Constant Weight Codes." IEEE Trans. Inform. Th. 36 , 1334-1380, 1990. Calderbank, A. R.; Hammons, A. R. Jr.; Kumar, P. V.; Sloane, N. J. A.; and Solé, P. "A Linear Construction for Certain Kerdock and Preparata Codes." Bull. Amer. Math. Soc. 29 , 218-222, 1993. Conway, J. H. and Sloane, N. J. A. "Quaternary Constructions for the Binary Single-Error-Correcting Codes of Julin, Best and Others." Des. Codes Cryptogr. 4 , 31-42, 1994. Conway, J. H. and Sloane, N. J. A. "Error-Correcting Codes." §3.2 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 75-88, 1993. Gachkov, I. "Error-Correcting Codes with Mathematica ." http://library.wolfram.com/infocenter/MathSource/5085/ . Gallian, J. "How Computers Can Read and Correct ID Numbers." Math Horizons , pp. 14-15, Winter 1993. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 119-121, 1994. MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-Correcting Codes. Amsterdam, Netherlands: North-Holland, 1977. Sloane, N. J. A. Sequences A000079 /M1129, A005864 /M1111, A005865 /M0240, and A005866 /M0226 in "The On-Line Encyclopedia of Integer Sequences." Sloane, N. J. A. and Plouffe, S. Figure M0240 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Referenced on Wolfram\|Alpha Error-Correcting Code Cite this as: Weisstein, Eric W. "Error-Correcting Code." From MathWorld --A Wolfram Resource. https://mathworld.wolfram.com/Error-CorrectingCode.html Subject classifications
Markdown	![](https://mathworld.wolfram.com/images/header/menu-icon.png) TOPICS ![](https://mathworld.wolfram.com/images/sidebar/menu-close.png) [Algebra](https://mathworld.wolfram.com/topics/Algebra.html) [Applied Mathematics](https://mathworld.wolfram.com/topics/AppliedMathematics.html) [Calculus and Analysis](https://mathworld.wolfram.com/topics/CalculusandAnalysis.html) [Discrete Mathematics](https://mathworld.wolfram.com/topics/DiscreteMathematics.html) [Foundations of Mathematics](https://mathworld.wolfram.com/topics/FoundationsofMathematics.html) [Geometry](https://mathworld.wolfram.com/topics/Geometry.html) [History and Terminology](https://mathworld.wolfram.com/topics/HistoryandTerminology.html) [Number Theory](https://mathworld.wolfram.com/topics/NumberTheory.html) [Probability and Statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html) [Recreational Mathematics](https://mathworld.wolfram.com/topics/RecreationalMathematics.html) [Topology](https://mathworld.wolfram.com/topics/Topology.html) [Alphabetical Index](https://mathworld.wolfram.com/letters/) [New in MathWorld](https://mathworld.wolfram.com/whatsnew/) - [Discrete Mathematics](https://mathworld.wolfram.com/topics/DiscreteMathematics.html) - [Coding Theory](https://mathworld.wolfram.com/topics/CodingTheory.html) # Error-Correcting Code * An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as [coding theory](https://mathworld.wolfram.com/CodingTheory.html). Error detection is much simpler than error correction, and one or more "check" digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes like Mariner used a type of error-correcting code called a block code, and more recent space probes use convolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modems use error detection when they compute [checksums](https://mathworld.wolfram.com/Checksum.html), which are sums of the digits in a given transmission modulo some number. The [ISBN](https://mathworld.wolfram.com/ISBN.html) used to identify books also incorporates a check [digit](https://mathworld.wolfram.com/Digit.html). A powerful check for 13 [digit](https://mathworld.wolfram.com/Digit.html) numbers consists of the following. Write the number as a string of [digits](https://mathworld.wolfram.com/Digit.html) ![a\_1a\_2a\_3...a\_(13)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline1.svg). Take ![a\_1+a\_3+...+a\_(13)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline2.svg) and double. Now add the number of [digits](https://mathworld.wolfram.com/Digit.html) in [odd](https://mathworld.wolfram.com/OddNumber.html) positions which are ![\>4](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline3.svg) to this number. Now add ![a\_2+a\_4+...+a\_(12)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline4.svg). The check number is then the number required to bring the last [digit](https://mathworld.wolfram.com/Digit.html) to 0. This scheme detects all single [digit](https://mathworld.wolfram.com/Digit.html) errors and all [transpositions](https://mathworld.wolfram.com/Transposition.html) of adjacent [digits](https://mathworld.wolfram.com/Digit.html) except 0 and 9. Let ![A(n,d)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline5.svg) denote the maximal number of ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline6.svg) (0,1)-vectors having the property that any two of the set differ in at least ![d](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline7.svg) places. The corresponding vectors can correct ![\[(d-1)/2\]](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline8.svg) errors. ![A(n,d,w)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline9.svg) is the number of ![A(n,d)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline10.svg)s with precisely ![w](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline11.svg) 1s (Sloane and Plouffe 1995). Since it is not possible for ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline12.svg)\-vectors to differ in ![d\>n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline13.svg) places and since ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline14.svg)\-vectors which differ in all ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline15.svg) places partition into disparate sets of two, \| \| \| \|---\|---\| \| ![ A(n,d)={1 n\<d; 2 n=d. ](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/NumberedEquation1.svg) \| (1) \| Values of ![A(n,d)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline16.svg) can be found by labeling the ![2^n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline17.svg) (0,1)-![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline18.svg)\-vectors, finding all unordered pairs ![(a\_i,a\_j)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline19.svg) of ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline20.svg)\-vectors which differ from each other in at least ![d](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline21.svg) places, forming a [graph](https://mathworld.wolfram.com/Graph.html) from these unordered pairs, and then finding the [clique number](https://mathworld.wolfram.com/CliqueNumber.html) of this graph. Unfortunately, finding the size of a clique for a given [graph](https://mathworld.wolfram.com/Graph.html) is an [NP-complete problem](https://mathworld.wolfram.com/NP-CompleteProblem.html). \| \| \| \| \|---\|---\|---\| \| ![d](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline22.svg) \| OEIS \| ![A(n,d)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline23.svg) \| \| 1 \| [A000079](http://oeis.org/A000079) \| 2, 4, 8, 16, 32, 64, 128, ... \| \| 2 \| \| 1, 2, 4, 8, ... \| \| 3 \| \| 1, 1, 2, 2, ... \| \| 4 \| [A005864](http://oeis.org/A005864) \| 1, 1, 1, 2, 4, 8, 16, 20, 40, ... \| \| 5 \| \| 1, 1, 1, 1, 2, ... \| \| 6 \| [A005865](http://oeis.org/A005865) \| 1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 12, ... \| \| 7 \| \| 1, 1, 1, 1, 1, 1, 2, ... \| \| 8 \| [A005866](http://oeis.org/A005866) \| 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, ... \| * ## See also [Checksum](https://mathworld.wolfram.com/Checksum.html), [Clique](https://mathworld.wolfram.com/Clique.html), [Clique Number](https://mathworld.wolfram.com/CliqueNumber.html), [Coding Theory](https://mathworld.wolfram.com/CodingTheory.html), [Finite Field](https://mathworld.wolfram.com/FiniteField.html), [Golay Code](https://mathworld.wolfram.com/GolayCode.html), [Hadamard Matrix](https://mathworld.wolfram.com/HadamardMatrix.html), [Halved Cube Graph](https://mathworld.wolfram.com/HalvedCubeGraph.html), [Hamming Code](https://mathworld.wolfram.com/HammingCode.html), [ISBN](https://mathworld.wolfram.com/ISBN.html), [Perfect Code](https://mathworld.wolfram.com/PerfectCode.html), [UPC](https://mathworld.wolfram.com/UPC.html) ## Explore with Wolfram\\|Alpha ![WolframAlpha](https://mathworld.wolfram.com/images/wolframalpha/WA-logo.png) More things to try: - [BCH code](https://www.wolframalpha.com/input/?i=BCH+code) - [coding theory](https://www.wolframalpha.com/input/?i=coding+theory) - [.999... = 1](https://www.wolframalpha.com/input/?i=.999...+%3D+1) ## References Baylis, J. [Error Correcting Codes: A Mathematical Introduction.](http://www.amazon.com/exec/obidos/ASIN/0412786907/ref=nosim/ericstreasuretro) Boca Raton, FL: CRC Press, 1998. Berlekamp, E. R. [Algebraic Coding Theory, rev. ed.](http://www.amazon.com/exec/obidos/ASIN/0894120638/ref=nosim/ericstreasuretro) New York: McGraw-Hill, 1968. Brouwer, A. E.; Shearer, J. B.; Sloane, N. J. A.; and Smith, W. D. "A New Table of Constant Weight Codes." IEEE Trans. Inform. Th. 36, 1334-1380, 1990. Calderbank, A. R.; Hammons, A. R. Jr.; Kumar, P. V.; Sloane, N. J. A.; and Solé, P. "A Linear Construction for Certain Kerdock and Preparata Codes." Bull. Amer. Math. Soc. 29, 218-222, 1993. Conway, J. H. and Sloane, N. J. A. "Quaternary Constructions for the Binary Single-Error-Correcting Codes of Julin, Best and Others." Des. Codes Cryptogr. 4, 31-42, 1994. Conway, J. H. and Sloane, N. J. A. "Error-Correcting Codes." §3.2 in [Sphere Packings, Lattices, and Groups, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/0387979123/ref=nosim/ericstreasuretro) New York: Springer-Verlag, pp. 75-88, 1993. ![](https://mathworld.wolfram.com/images/entries/mathematica.gif) Gachkov, I. "Error-Correcting Codes with Mathematica." <http://library.wolfram.com/infocenter/MathSource/5085/>. Gallian, J. "How Computers Can Read and Correct ID Numbers." Math Horizons, pp. 14-15, Winter 1993. Guy, R. K. [Unsolved Problems in Number Theory, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/0387208607/ref=nosim/ericstreasuretro) New York: Springer-Verlag, pp. 119-121, 1994. MacWilliams, F. J. and Sloane, N. J. A. [The Theory of Error-Correcting Codes.](http://www.amazon.com/exec/obidos/ASIN/0444851933/ref=nosim/ericstreasuretro) Amsterdam, Netherlands: North-Holland, 1977. Sloane, N. J. A. Sequences [A000079](http://oeis.org/A000079)/M1129, [A005864](http://oeis.org/A005864)/M1111, [A005865](http://oeis.org/A005865)/M0240, and [A005866](http://oeis.org/A005866)/M0226 in "The On-Line Encyclopedia of Integer Sequences." Sloane, N. J. A. and Plouffe, S. Figure M0240 in [The Encyclopedia of Integer Sequences.](http://www.amazon.com/exec/obidos/ASIN/0125586302/ref=nosim/ericstreasuretro) San Diego: Academic Press, 1995. ## Referenced on Wolfram\\|Alpha [Error-Correcting Code](https://www.wolframalpha.com/input/?i=error-correcting+code "Error-Correcting Code") ## Cite this as: [Weisstein, Eric W.](https://mathworld.wolfram.com/about/author.html) "Error-Correcting Code." From [MathWorld](https://mathworld.wolfram.com/)\--A Wolfram Resource. <https://mathworld.wolfram.com/Error-CorrectingCode.html> ## Subject classifications - [Discrete Mathematics](https://mathworld.wolfram.com/topics/DiscreteMathematics.html) - [Coding Theory](https://mathworld.wolfram.com/topics/CodingTheory.html) - [About MathWorld](https://mathworld.wolfram.com/about/) - [MathWorld Classroom](https://mathworld.wolfram.com/classroom/) - [Contribute](https://mathworld.wolfram.com/contact/) - [MathWorld Book](https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20) - [wolfram.com](https://www.wolfram.com/) - [13,311 Entries](https://mathworld.wolfram.com/whatsnew/) - [Last Updated: Wed Mar 25 2026](https://mathworld.wolfram.com/whatsnew/) - [©1999–2026 Wolfram Research, Inc.](https://www.wolfram.com/) - [Terms of Use](https://www.wolfram.com/legal/terms/mathworld.html) - [![Wolfram](https://mathworld.wolfram.com/images/footer/wolfram-logo.png)](https://www.wolfram.com/) - [wolfram.com](https://www.wolfram.com/) - [Wolfram for Education](https://www.wolfram.com/education/) - Created, developed and nurtured by Eric Weisstein at Wolfram Research Created, developed and nurtured by Eric Weisstein at Wolfram Research [Find out if you already have access to Wolfram tech through your organization](https://www.wolfram.com/siteinfo/) ×
Readable Markdown	[Algebra](https://mathworld.wolfram.com/topics/Algebra.html) [Applied Mathematics](https://mathworld.wolfram.com/topics/AppliedMathematics.html) [Calculus and Analysis](https://mathworld.wolfram.com/topics/CalculusandAnalysis.html) [Discrete Mathematics](https://mathworld.wolfram.com/topics/DiscreteMathematics.html) [Foundations of Mathematics](https://mathworld.wolfram.com/topics/FoundationsofMathematics.html) [Geometry](https://mathworld.wolfram.com/topics/Geometry.html) [History and Terminology](https://mathworld.wolfram.com/topics/HistoryandTerminology.html) [Number Theory](https://mathworld.wolfram.com/topics/NumberTheory.html) [Probability and Statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html) [Recreational Mathematics](https://mathworld.wolfram.com/topics/RecreationalMathematics.html) [Topology](https://mathworld.wolfram.com/topics/Topology.html) [Alphabetical Index](https://mathworld.wolfram.com/letters/) [New in MathWorld](https://mathworld.wolfram.com/whatsnew/) * An error-correcting code is an algorithm for expressing a sequence of numbers such that any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers. The study of error-correcting codes and the associated mathematics is known as [coding theory](https://mathworld.wolfram.com/CodingTheory.html). Error detection is much simpler than error correction, and one or more "check" digits are commonly embedded in credit card numbers in order to detect mistakes. Early space probes like Mariner used a type of error-correcting code called a block code, and more recent space probes use convolution codes. Error-correcting codes are also used in CD players, high speed modems, and cellular phones. Modems use error detection when they compute [checksums](https://mathworld.wolfram.com/Checksum.html), which are sums of the digits in a given transmission modulo some number. The [ISBN](https://mathworld.wolfram.com/ISBN.html) used to identify books also incorporates a check [digit](https://mathworld.wolfram.com/Digit.html). A powerful check for 13 [digit](https://mathworld.wolfram.com/Digit.html) numbers consists of the following. Write the number as a string of [digits](https://mathworld.wolfram.com/Digit.html) ![a\_1a\_2a\_3...a\_(13)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline1.svg). Take ![a\_1+a\_3+...+a\_(13)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline2.svg) and double. Now add the number of [digits](https://mathworld.wolfram.com/Digit.html) in [odd](https://mathworld.wolfram.com/OddNumber.html) positions which are ![\>4](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline3.svg) to this number. Now add ![a\_2+a\_4+...+a\_(12)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline4.svg). The check number is then the number required to bring the last [digit](https://mathworld.wolfram.com/Digit.html) to 0. This scheme detects all single [digit](https://mathworld.wolfram.com/Digit.html) errors and all [transpositions](https://mathworld.wolfram.com/Transposition.html) of adjacent [digits](https://mathworld.wolfram.com/Digit.html) except 0 and 9. Let ![A(n,d)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline5.svg) denote the maximal number of ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline6.svg) (0,1)-vectors having the property that any two of the set differ in at least ![d](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline7.svg) places. The corresponding vectors can correct ![\[(d-1)/2\]](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline8.svg) errors. ![A(n,d,w)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline9.svg) is the number of ![A(n,d)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline10.svg)s with precisely ![w](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline11.svg) 1s (Sloane and Plouffe 1995). Since it is not possible for ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline12.svg)\-vectors to differ in ![d\>n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline13.svg) places and since ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline14.svg)\-vectors which differ in all ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline15.svg) places partition into disparate sets of two, \| \| \| \|---\|---\| \| ![ A(n,d)={1 n\<d; 2 n=d. ](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/NumberedEquation1.svg) \| (1) \| Values of ![A(n,d)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline16.svg) can be found by labeling the ![2^n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline17.svg) (0,1)-![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline18.svg)\-vectors, finding all unordered pairs ![(a\_i,a\_j)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline19.svg) of ![n](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline20.svg)\-vectors which differ from each other in at least ![d](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline21.svg) places, forming a [graph](https://mathworld.wolfram.com/Graph.html) from these unordered pairs, and then finding the [clique number](https://mathworld.wolfram.com/CliqueNumber.html) of this graph. Unfortunately, finding the size of a clique for a given [graph](https://mathworld.wolfram.com/Graph.html) is an [NP-complete problem](https://mathworld.wolfram.com/NP-CompleteProblem.html). \| \| \| \| \|---\|---\|---\| \| ![d](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline22.svg) \| OEIS \| ![A(n,d)](https://mathworld.wolfram.com/images/equations/Error-CorrectingCode/Inline23.svg) \| \| 1 \| [A000079](http://oeis.org/A000079) \| 2, 4, 8, 16, 32, 64, 128, ... \| \| 2 \| \| 1, 2, 4, 8, ... \| \| 3 \| \| 1, 1, 2, 2, ... \| \| 4 \| [A005864](http://oeis.org/A005864) \| 1, 1, 1, 2, 4, 8, 16, 20, 40, ... \| \| 5 \| \| 1, 1, 1, 1, 2, ... \| \| 6 \| [A005865](http://oeis.org/A005865) \| 1, 1, 1, 1, 1, 2, 2, 2, 4, 6, 12, ... \| \| 7 \| \| 1, 1, 1, 1, 1, 1, 2, ... \| \| 8 \| [A005866](http://oeis.org/A005866) \| 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, ... \| * ## See also [Checksum](https://mathworld.wolfram.com/Checksum.html), [Clique](https://mathworld.wolfram.com/Clique.html), [Clique Number](https://mathworld.wolfram.com/CliqueNumber.html), [Coding Theory](https://mathworld.wolfram.com/CodingTheory.html), [Finite Field](https://mathworld.wolfram.com/FiniteField.html), [Golay Code](https://mathworld.wolfram.com/GolayCode.html), [Hadamard Matrix](https://mathworld.wolfram.com/HadamardMatrix.html), [Halved Cube Graph](https://mathworld.wolfram.com/HalvedCubeGraph.html), [Hamming Code](https://mathworld.wolfram.com/HammingCode.html), [ISBN](https://mathworld.wolfram.com/ISBN.html), [Perfect Code](https://mathworld.wolfram.com/PerfectCode.html), [UPC](https://mathworld.wolfram.com/UPC.html) ## Explore with Wolfram\\|Alpha ## References Baylis, J. [Error Correcting Codes: A Mathematical Introduction.](http://www.amazon.com/exec/obidos/ASIN/0412786907/ref=nosim/ericstreasuretro) Boca Raton, FL: CRC Press, 1998.Berlekamp, E. R. [Algebraic Coding Theory, rev. ed.](http://www.amazon.com/exec/obidos/ASIN/0894120638/ref=nosim/ericstreasuretro) New York: McGraw-Hill, 1968.Brouwer, A. E.; Shearer, J. B.; Sloane, N. J. A.; and Smith, W. D. "A New Table of Constant Weight Codes." IEEE Trans. Inform. Th. 36, 1334-1380, 1990.Calderbank, A. R.; Hammons, A. R. Jr.; Kumar, P. V.; Sloane, N. J. A.; and Solé, P. "A Linear Construction for Certain Kerdock and Preparata Codes." Bull. Amer. Math. Soc. 29, 218-222, 1993.Conway, J. H. and Sloane, N. J. A. "Quaternary Constructions for the Binary Single-Error-Correcting Codes of Julin, Best and Others." Des. Codes Cryptogr. 4, 31-42, 1994.Conway, J. H. and Sloane, N. J. A. "Error-Correcting Codes." §3.2 in [Sphere Packings, Lattices, and Groups, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/0387979123/ref=nosim/ericstreasuretro) New York: Springer-Verlag, pp. 75-88, 1993.![](https://mathworld.wolfram.com/images/entries/mathematica.gif) Gachkov, I. "Error-Correcting Codes with Mathematica." <http://library.wolfram.com/infocenter/MathSource/5085/>.Gallian, J. "How Computers Can Read and Correct ID Numbers." Math Horizons, pp. 14-15, Winter 1993.Guy, R. K. [Unsolved Problems in Number Theory, 2nd ed.](http://www.amazon.com/exec/obidos/ASIN/0387208607/ref=nosim/ericstreasuretro) New York: Springer-Verlag, pp. 119-121, 1994.MacWilliams, F. J. and Sloane, N. J. A. [The Theory of Error-Correcting Codes.](http://www.amazon.com/exec/obidos/ASIN/0444851933/ref=nosim/ericstreasuretro) Amsterdam, Netherlands: North-Holland, 1977.Sloane, N. J. A. Sequences [A000079](http://oeis.org/A000079)/M1129, [A005864](http://oeis.org/A005864)/M1111, [A005865](http://oeis.org/A005865)/M0240, and [A005866](http://oeis.org/A005866)/M0226 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. Figure M0240 in [The Encyclopedia of Integer Sequences.](http://www.amazon.com/exec/obidos/ASIN/0125586302/ref=nosim/ericstreasuretro) San Diego: Academic Press, 1995. ## Referenced on Wolfram\\|Alpha [Error-Correcting Code](https://www.wolframalpha.com/input/?i=error-correcting+code "Error-Correcting Code") ## Cite this as: [Weisstein, Eric W.](https://mathworld.wolfram.com/about/author.html) "Error-Correcting Code." From [MathWorld](https://mathworld.wolfram.com/)\--A Wolfram Resource. <https://mathworld.wolfram.com/Error-CorrectingCode.html> ## Subject classifications
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