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| Boilerpipe Text | "Interleaver" redirects here. For the fiber-optic device, see
optical interleaver
.
In
computing
,
telecommunication
,
information theory
, and
coding theory
,
forward error correction
(
FEC
) or
channel coding
[
1
]
is a technique used for
controlling errors
in
data transmission
over unreliable or noisy
communication channels
.
The central idea is that the sender encodes the message in a
redundant
way, most often by using an
error correction code
, or
error correcting code
(
ECC
).
[
2
]
[
3
]
The redundancy allows the receiver not only to
detect errors
that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a
reverse channel
to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth.
The American mathematician
Richard Hamming
pioneered this field in the 1940s and invented the first error-correcting code in 1950: the
Hamming (7,4) code
.
[
3
]
FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in
multicast
.
Long-latency connections also benefit; in the case of satellites orbiting distant planets, retransmission due to errors would create a delay of several hours. FEC is also widely used in
modems
and in
cellular networks
.
FEC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, FEC is an integral part of the initial
analog-to-digital conversion
in the receiver. The
Viterbi decoder
implements a
soft-decision algorithm
to demodulate digital data from an analog signal corrupted by noise. Many FEC decoders can also generate a
bit-error rate
(BER) signal which can be used as feedback to fine-tune the analog receiving electronics.
FEC information is added to
mass storage
(magnetic, optical and solid state/flash based) devices to enable recovery of corrupted data, and is used as
ECC
computer memory
on systems that require special provisions for reliability.
The maximum proportion of errors or missing bits that can be corrected is determined by the design of the ECC, so different forward error correcting codes are suitable for different conditions. In general, a stronger code induces more redundancy that needs to be transmitted using the available bandwidth, which reduces the effective bit-rate while improving the received effective
signal-to-noise ratio
. The
noisy-channel coding theorem
of
Claude Shannon
can be used to compute the maximum achievable communication bandwidth for a given maximum acceptable error probability. This establishes bounds on the theoretical maximum information transfer rate of a channel with some given base noise level. However, the proof is not constructive, and hence gives no insight of how to build a capacity achieving code. After years of research, some advanced FEC systems like
polar code
[
4
]
come very close to the theoretical maximum given by the Shannon channel capacity under the hypothesis of an infinite length frame.
ECC is accomplished by adding
redundancy
to the transmitted information using an algorithm. A redundant bit may be a complicated function of many original information bits. The original information may or may not appear literally in the encoded output; codes that include the unmodified input in the output are
systematic
, while those that do not are
non-systematic
.
A simplistic example of ECC is to transmit each data bit three times, which is known as a (3,1)
repetition code
. Through a noisy channel, a receiver might see eight versions of the output; see the table below.
Triplet received
Interpreted as
000
0 (error-free)
001
0
010
0
100
0
111
1 (error-free)
110
1
101
1
011
1
This allows an error in any one of the three samples to be corrected by "majority vote" or "democratic voting". The correcting ability of this ECC is:
up to one bit of triplet in error, or
up to two bits of triplet omitted (cases not shown in table).
Though simple to implement and widely used, this
triple modular redundancy
is a relatively inefficient ECC. Better ECC codes typically examine the last several tens or even the last several hundreds of previously received bits to determine how to decode the current small handful of bits (typically in groups of two to eight bits).
Simplified formalism
[
edit
]
Formally, an error-correcting code is given by its (
injective
) encoding function
which assigns to each word
of a finite
alphabet
a unique word
(a concatenation of letters) from the
alphabet
.
Most commonly,
is a
homomorphism
in the sense that if
is the concatenation of
and
, then we have the following:
This implies that it is enough to define
for single-letter words
. The
range
of the function
is the set of
code-words.
The capabilities of the code to detect and correct errors can then be understood from the
distance
of the code, which is the minimum
Hamming distance
separating any two distinct code words. A code with distance
can detect errors on
bits as long as
, and among those detected errors, the code can correct
-bit errors whenever
.
Averaging noise to reduce errors
[
edit
]
ECC could be said to work by "averaging noise"; since each data bit affects many transmitted symbols, the corruption of some symbols by noise usually allows the original user data to be extracted from the other, uncorrupted received symbols that also depend on the same user data.
Because of this "risk-pooling" effect, digital communication systems that use ECC tend to work well above a certain minimum
signal-to-noise ratio
and not at all below it.
This
all-or-nothing tendency
ā the
cliff effect
ā becomes more pronounced as stronger codes are used that more closely approach the theoretical
Shannon limit
.
Interleaving ECC coded data can reduce the all or nothing properties of transmitted ECC codes when the channel errors tend to occur in bursts. However, this method has limits; it is best used on narrowband data.
Most telecommunication systems use a fixed
channel code
designed to tolerate the expected worst-case
bit error rate
, and then fail to work at all if the bit error rate is ever worse.
However, some systems adapt to the given channel error conditions: some instances of
hybrid automatic repeat-request
use a fixed ECC method as long as the ECC can handle the error rate, then switch to
ARQ
when the error rate gets too high;
adaptive modulation and coding
uses a variety of ECC rates, adding more error-correction bits per packet when there are higher error rates in the channel, or taking them out when they are not needed.
A block code (specifically a
Hamming code
) where redundant bits are added as a block to the end of the initial message
A continuous
convolutional code
where redundant bits are added continuously into the structure of the code word
The two main categories of ECC codes are
block codes
and
convolutional codes
.
Block codes work on fixed-size blocks (packets) of bits or symbols of predetermined size. Practical block codes can generally be hard-decoded in
polynomial time
to their block length.
Convolutional codes work on bit or symbol streams of arbitrary length. They are most often soft decoded with the
Viterbi algorithm
, though other algorithms are sometimes used. Viterbi decoding allows asymptotically optimal decoding efficiency with increasing constraint length of the convolutional code, but at the expense of
exponentially
increasing complexity. A convolutional code that is terminated is also a 'block code' in that it encodes a block of input data, but the block size of a convolutional code is generally arbitrary, while block codes have a fixed size dictated by their algebraic characteristics. Types of termination for convolutional codes include "tail-biting" and "bit-flushing".
Classical block codes are usually decoded using
hard-decision
algorithms,
[
5
]
which means that for every input and output signal a hard decision is made whether it corresponds to a one or a zero bit. In contrast, convolutional codes are typically decoded using
soft-decision
algorithms like the Viterbi, MAP or
BCJR
algorithms, which process (discretized) analog signals, and which allow for much higher error-correction performance than hard-decision decoding.
Nearly all classical block codes apply the algebraic properties of
finite fields
. Hence classical block codes are often referred to as algebraic codes.
There are many types of block codes;
ReedāSolomon coding
is noteworthy for its widespread use in
compact discs
,
DVDs
, and
hard disk drives
. Other examples of classical block codes include
Golay
,
BCH
,
Multidimensional parity
, and
Hamming codes
.
Hamming ECC is commonly used to correct
ECC memory
and early SLC
NAND flash
memory errors.
[
6
]
This provides single-bit error correction and 2-bit error detection.
Hamming codes are only suitable for more reliable
single-level cell
(SLC) NAND.
Denser
multi-level cell
(MLC) NAND may use multi-bit correcting ECC such as
BCH
, ReedāSolomon, or
LDPC
.
[
7
]
[
8
]
NOR flash typically does not use any error correction.
[
7
]
Low-density parity-check (LDPC)
[
edit
]
Low-density parity-check
(LDPC) codes are a class of highly efficient linear block
codes made from many single parity check (SPC) codes. They can provide performance very close to the
channel capacity
(the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length. Practical implementations rely heavily on decoding the constituent SPC codes in parallel.
LDPC codes were first introduced by
Robert G. Gallager
in his PhD thesis in 1960,
but due to the computational effort in implementing encoder and decoder and the introduction of
ReedāSolomon
codes,
they were mostly ignored until the 1990s.
LDPC codes are now used in many recent high-speed communication standards, such as
DVB-S2
(Digital Video Broadcasting ā Satellite ā Second Generation),
WiMAX
(
IEEE 802.16e
standard for microwave communications), High-Speed Wireless LAN (
IEEE 802.11n
),
[
9
]
10GBase-T Ethernet
(802.3an) and
G.hn/G.9960
(ITU-T Standard for networking over power lines, phone lines and coaxial cable). Other LDPC codes are standardized for wireless communication standards within
3GPP
MBMS
(see
fountain codes
).
Turbo coding
is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the
Shannon limit
. Predating
LDPC codes
in terms of practical application, they now provide similar performance.
One of the earliest commercial applications of turbo coding was the
CDMA2000 1x
(TIA IS-2000) digital cellular technology developed by
Qualcomm
and sold by
Verizon Wireless
,
Sprint
, and other carriers. It is also used for the evolution of CDMA2000 1x specifically for Internet access,
1xEV-DO
(TIA IS-856). Like 1x, EV-DO was developed by
Qualcomm
, and is sold by
Verizon Wireless
,
Sprint
, and other carriers (Verizon's marketing name for 1xEV-DO is
Broadband Access
, Sprint's consumer and business marketing names for 1xEV-DO are
Power Vision
and
Mobile Broadband
, respectively).
Describing the performance of an ECC
[
edit
]
In contrast to classical block codes that often specify an error-detecting or error-correcting ability, many modern block codes such as
LDPC codes
lack such guarantees. Instead, modern codes are evaluated in terms of their bit error rates.
Most forward error correction codes correct only bit-flips, but not bit-insertions or bit-deletions.
In this setting, the
Hamming distance
is the appropriate way to measure the
bit error rate
.
A few forward error correction codes are designed to correct bit-insertions and bit-deletions, such as Marker Codes and Watermark Codes.
The
Levenshtein distance
is a more appropriate way to measure the bit error rate when using such codes.
[
10
]
Code-rate and the tradeoff between reliability and data rate
[
edit
]
The fundamental principle of ECC is to add redundant bits in order to help the decoder to find out the true message that was encoded by the transmitter. The code-rate of a given ECC system is defined as the ratio between the number of information bits and the total number of bits (i.e., information plus redundancy bits) in a given communication package. The code-rate is hence a real number. A low code-rate close to zero implies a strong code that uses many redundant bits to achieve a good performance, while a large code-rate close to 1 implies a weak code.
The redundant bits that protect the information have to be transferred using the same communication resources that they are trying to protect. This causes a fundamental tradeoff between reliability and data rate.
[
11
]
In one extreme, a strong code (with low code-rate) can induce an important increase in the receiver SNR (signal-to-noise-ratio) decreasing the bit error rate, at the cost of reducing the effective data rate. On the other extreme, not using any ECC (i.e., a code-rate equal to 1) uses the full channel for information transfer purposes, at the cost of leaving the bits without any additional protection.
One interesting question is the following: how efficient in terms of information transfer can an ECC be that has a negligible decoding error rate? This question was answered by Claude Shannon with his second theorem, which says that the channel capacity is the maximum bit rate achievable by any ECC whose error rate tends to zero:
[
12
]
His proof relies on Gaussian random coding, which is not suitable to real-world applications. The upper bound given by Shannon's work inspired a long journey in designing ECCs that can come close to the ultimate performance boundary. Various codes today can attain almost the Shannon limit. However, capacity achieving ECCs are usually extremely complex to implement.
The most popular ECCs have a trade-off between performance and computational complexity. Usually, their parameters give a range of possible code rates, which can be optimized depending on the scenario. Usually, this optimization is done in order to achieve a low decoding error probability while minimizing the impact to the data rate. Another criterion for optimizing the code rate is to balance low error rate and retransmissions number in order to the energy cost of the communication.
[
13
]
Local decoding and testing of codes
[
edit
]
Sometimes it is only necessary to decode single bits of the message, or to check whether a given signal is a codeword, and do so without looking at the entire signal. This can make sense in a streaming setting, where codewords are too large to be classically decoded fast enough and where only a few bits of the message are of interest for now. Also such codes have become an important tool in
computational complexity theory
, e.g., for the design of
probabilistically checkable proofs
.
Locally decodable codes
are error-correcting codes for which single bits of the message can be probabilistically recovered by only looking at a small (say constant) number of positions of a codeword, even after the codeword has been corrupted at some constant fraction of positions.
Locally testable codes
are error-correcting codes for which it can be checked probabilistically whether a signal is close to a codeword by only looking at a small number of positions of the signal.
Not all locally decodable codes (LDCs) are locally testable codes (LTCs)
[
14
]
neither locally correctable codes (LCCs),
[
15
]
q-query LCCs are bounded exponentially
[
16
]
while LDCs can have
subexponential
lengths.
[
17
]
Improving performance
[
edit
]
Concatenation (combination)
[
edit
]
Classical (algebraic) block codes and convolutional codes are frequently combined in
concatenated
coding schemes in which a short constraint-length Viterbi-decoded convolutional code does most of the work and a block code (usually ReedāSolomon) with larger symbol size and block length "mops up" any errors made by the convolutional decoder. Single pass decoding with this family of error correction codes can yield very low error rates, but for long range transmission conditions (like deep space) iterative decoding is recommended.
Concatenated codes have been standard practice in satellite and deep space communications since
Voyager 2
first used the technique in its 1986 encounter with
Uranus
. The
Galileo
craft used iterative concatenated codes to compensate for the very high error rate conditions caused by having a failed antenna.
"Interleaver" redirects here. For the fiber-optic device, see
optical interleaver
.
A short illustration of the interleaving idea
Interleaving is frequently used in digital communication and storage systems to improve the performance of forward error correcting codes. Many
communication channels
are not memoryless: errors typically occur in
bursts
rather than independently. If the number of errors within a
code word
exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving alleviates this problem by shuffling source symbols across several code words, thereby creating a more
uniform distribution
of errors.
[
18
]
Therefore, interleaving is widely used for
burst error-correction
.
The analysis of modern iterated codes, like
turbo codes
and
LDPC codes
, typically assumes an independent distribution of errors.
[
19
]
Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word.
[
20
]
For turbo codes, an interleaver is an integral component and its proper design is crucial for good performance.
[
18
]
[
21
]
The iterative decoding algorithm works best when there are not short cycles in the
factor graph
that represents the decoder; the interleaver is chosen to avoid short cycles.
Interleaver designs include:
rectangular (or uniform) interleavers (similar to the method using skip factors described above)
convolutional interleavers
random interleavers (where the interleaver is a known random permutation)
S-random interleaver (where the interleaver is a known random permutation with the constraint that no input symbols within distance S appear within a distance of S in the output).
[
22
]
a contention-free quadratic
permutation polynomial
(QPP).
[
23
]
An example of use is in the
3GPP Long Term Evolution
mobile telecommunication standard.
[
24
]
In multi-
carrier
communication systems, interleaving across carriers may be employed to provide frequency
diversity
, e.g., to mitigate
frequency-selective fading
or narrowband interference.
[
25
]
Interleaving example
[
edit
]
Transmission without interleaving
:
Error-free message: aaaabbbbccccddddeeeeffffgggg
Transmission with a burst error: aaaabbbbccc____deeeeffffgggg
Here, each group of the same letter represents a 4-bit one-bit error-correcting codeword. The codeword cccc is altered in one bit and can be corrected, but the codeword dddd is altered in three bits, so either it cannot be decoded at all or it might be
decoded incorrectly
.
With interleaving
:
Error-free code words: aaaabbbbccccddddeeeeffffgggg
Interleaved: abcdefgabcdefgabcdefgabcdefg
Transmission with a burst error: abcdefgabcd____bcdefgabcdefg
Received code words after deinterleaving: aa_abbbbccccdddde_eef_ffg_gg
In each of the codewords "aaaa", "eeee", "ffff", and "gggg", only one bit is altered, so one-bit error-correcting code will decode everything correctly.
Transmission without interleaving
:
Original transmitted sentence: ThisIsAnExampleOfInterleaving
Received sentence with a burst error: ThisIs______pleOfInterleaving
The term "AnExample" ends up mostly unintelligible and difficult to correct.
With interleaving
:
Transmitted sentence: ThisIsAnExampleOfInterleaving...
Error-free transmission: TIEpfeaghsxlIrv.iAaenli.snmOten.
Received sentence with a burst error: TIEpfe______Irv.iAaenli.snmOten.
Received sentence after deinterleaving: T_isI_AnE_amp_eOfInterle_vin_...
No word is completely lost and the missing letters can be recovered with minimal guesswork.
Disadvantages of interleaving
[
edit
]
Use of interleaving techniques increases total delay. This is because the entire interleaved block must be received before the packets can be decoded.
[
26
]
Also interleavers hide the structure of errors; without an interleaver, more advanced decoding algorithms can take advantage of the error structure and achieve more reliable communication than a simpler decoder combined with an interleaver
[
citation needed
]
. An example of such an algorithm is based on
neural network
[
27
]
structures.
Software for error-correcting codes
[
edit
]
Simulating the behaviour of error-correcting codes (ECCs) in software is a common practice to design, validate and improve ECCs. The upcoming wireless 5G standard raises a new range of applications for the software ECCs: the
Cloud Radio Access Networks (C-RAN)
in a
Software-defined radio (SDR)
context. The idea is to directly use software ECCs in the communications. For instance in the 5G, the software ECCs could be located in the cloud and the antennas connected to this computing resources: improving this way the flexibility of the communication network and eventually increasing the energy efficiency of the system.
In this context, there are various available Open-source software listed below (non exhaustive).
AFF3CT
(A Fast Forward Error Correction Toolbox): a full communication chain in C++ (many supported codes like Turbo, LDPC, Polar codes, etc.), very fast and specialized on channel coding (can be used as a program for simulations or as a library for the SDR).
IT++
: a C++ library of classes and functions for linear algebra, numerical optimization, signal processing, communications, and statistics.
OpenAir
: implementation (in C) of the 3GPP specifications concerning the Evolved Packet Core Networks.
List of error-correcting codes
[
edit
]
Hard codes
Code
Distance
Detectable errors (bits)
Correctable errors (bits)
Parity (guess needed on error)
2
1
0
Triple modular redundancy
3
2
1
Perfect Hamming such as
Hamming(7,4)
3
2
1
SECDED
: extended Hamming such as (39,32), (72,64)
4
3
1
DECTED:
Nordstrom-Robinson code
6
5
2
Perfect
binary Golay code
7
6
3
TECFED: Extended
binary Golay code
8
7
3
In the table above, for an error-correcting code of minimal Hamming distance
, the maximum number of errors that the code can detect is given by
while the maximal number of errors that it can correct is given by
.
AN codes
Algebraic geometry code
BCH code
, which can be designed to correct any arbitrary number of errors per code block.
Barker code
used for radar, telemetry, ultra sound, Wifi, DSSS mobile phone networks, GPS etc.
Berger code
Constant-weight code
Convolutional code
Expander codes
Group codes
Golay codes
, of which the
Binary Golay code
is of practical interest
Goppa code
, used in the
McEliece cryptosystem
Hadamard code
Hagelbarger code
Hamming code
Latin square based code
for non-white noise (prevalent for example in broadband over powerlines)
Lexicographic code
Linear Network Coding
, a type of erasure correcting code across networks instead of point-to-point links
Long code
Low-density parity-check code
, also known as
Gallager code
, as the archetype for
sparse graph codes
LT code
, which is a near-optimal
rateless erasure correcting code (Fountain code)
m of n codes
Nordstrom-Robinson code
, used in Geometry and Group Theory
[
28
]
Online code
, a near-optimal
rateless erasure correcting code
Polar code (coding theory)
Raptor code
, a near-optimal
rateless erasure correcting code
ReedāSolomon error correction
ReedāMuller code
Repeat-accumulate code
Repetition codes
, such as
Triple modular redundancy
Spinal code, a rateless, nonlinear code based on pseudo-random hash functions
[
29
]
Tornado code
, a near-optimal
erasure correcting code
, and the precursor to
Fountain codes
Turbo code
WalshāHadamard code
Cyclic redundancy checks
(CRCs) can correct 1-bit errors for messages at most
bits long for optimal generator polynomials of degree
, see
Mathematics of cyclic redundancy checks Ā§Ā Bitfilters
Locally Recoverable Codes
Message authentication code
Burst error-correcting code
Code rate
Erasure codes
Error detection and correction
Error-correcting codes with feedback
Linear code
Quantum error correction
Soft-decision decoder
^
Charles Wang; Dean Sklar; Diana Johnson (Winter 2001ā2002).
"Forward Error-Correction Coding"
.
Crosslink
.
3
(1). The Aerospace Corporation. Archived from
the original
on 14 March 2012
. Retrieved
5 March
2006
.
^
Glover, Neal; Dudley, Trent (1990).
Practical Error Correction Design For Engineers
(Revision 1.1, 2ndĀ ed.). CO, USA:
Cirrus Logic
.
ISBN
Ā
0-927239-00-0
.
^
a
b
Hamming, Richard Wesley
(April 1950). "Error Detecting and Error Correcting Codes".
Bell System Technical Journal
.
29
(2). USA:
AT&T
:
147ā
160.
Bibcode
:
1950BSTJ...29..147H
.
doi
:
10.1002/j.1538-7305.1950.tb00463.x
.
hdl
:
10945/46756
.
S2CID
Ā
61141773
.
^
Maunder, Robert (2016).
"Overview of Channel Coding"
.
^
Baldi, M.; Chiaraluce, F. (2008).
"A Simple Scheme for Belief Propagation Decoding of BCH and RS Codes in Multimedia Transmissions"
.
International Journal of Digital Multimedia Broadcasting
.
2008
:
1ā
12.
doi
:
10.1155/2008/957846
.
^
"Hamming codes for NAND flash memory devices"
Archived
21 August 2016 at the
Wayback Machine
. EE Times-Asia. Apparently based on
"Micron Technical Note TN-29-08: Hamming Codes for NAND Flash Memory Devices"
Archived
29 August 2017 at the
Wayback Machine
. 2005. Both say: "The Hamming algorithm is an industry-accepted method for error detection and correction in many SLC NAND flash-based applications."
^
a
b
"What Types of ECC Should Be Used on Flash Memory?"
(PDF)
. Spansion. 2011. Archived from
the original
(Application note)
on 14 June 2016.
Both ReedāSolomon algorithm and BCH algorithm are common ECC choices for MLC NAND flash. ... Hamming based block codes are the most commonly used ECC for SLC.... both ReedāSolomon and BCH are able to handle multiple errors and are widely used on MLC flash.
^
Jim Cooke (August 2007).
"The Inconvenient Truths of NAND Flash Memory"
(PDF)
. p.Ā 28.
For SLC, a code with a correction threshold of 1 is sufficient. t=4 required ... for MLC.
^
IEEE Standard, section 20.3.11.6
"802.11n-2009"
Archived
3 February 2013 at the
Wayback Machine
, IEEE, 29 October 2009, accessed 21 March 2011.
^
Shah, Gaurav; Molina, Andres; Blaze, Matt (2006).
"Keyboards and covert channels"
.
USENIX
. Retrieved
20 December
2018
.
^
Tse, David; Viswanath, Pramod (2005),
Fundamentals of Wireless Communication
,
Cambridge University Press
, UK
^
Shannon, C. E. (1948).
"A mathematical theory of communication"
(PDF)
.
Bell System Technical Journal
.
27
(
3ā
4): 379ā423 & 623ā656.
Bibcode
:
1948BSTJ...27..379S
.
doi
:
10.1002/j.1538-7305.1948.tb01338.x
.
hdl
:
11858/00-001M-0000-002C-4314-2
.
^
Rosas, F.; Brante, G.; Souza, R. D.; Oberli, C. (2014). "Optimizing the code rate for achieving energy-efficient wireless communications".
Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC)
. pp.Ā
775ā
780.
doi
:
10.1109/WCNC.2014.6952166
.
ISBN
Ā
978-1-4799-3083-8
.
^
Kaufman, Tali
; Viderman, Michael.
"Locally Testable vs. Locally Decodable Codes"
.
^
Brubaker, Ben (9 January 2024).
"
'Magical' Error Correction Scheme Proved Inherently Inefficient"
.
Quanta Magazine
. Retrieved
9 January
2024
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error correction zoo
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lpdec: library for LP decoding and related things (Python) |
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## Contents
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- [1 Method](https://en.wikipedia.org/wiki/Error_correction_code#Method)
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- [1\.1 Simplified formalism](https://en.wikipedia.org/wiki/Error_correction_code#Simplified_formalism)
- [1\.2 Averaging noise to reduce errors](https://en.wikipedia.org/wiki/Error_correction_code#Averaging_noise_to_reduce_errors)
- [2 Types](https://en.wikipedia.org/wiki/Error_correction_code#Types)
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- [2\.1 Block codes](https://en.wikipedia.org/wiki/Error_correction_code#Block_codes)
- [2\.2 Soft codes](https://en.wikipedia.org/wiki/Error_correction_code#Soft_codes)
- [2\.2.1 Low-density parity-check (LDPC)](https://en.wikipedia.org/wiki/Error_correction_code#Low-density_parity-check_\(LDPC\))
- [2\.2.2 Turbo code](https://en.wikipedia.org/wiki/Error_correction_code#Turbo_code)
- [3 Describing the performance of an ECC](https://en.wikipedia.org/wiki/Error_correction_code#Describing_the_performance_of_an_ECC)
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- [3\.1 Code-rate and the tradeoff between reliability and data rate](https://en.wikipedia.org/wiki/Error_correction_code#Code-rate_and_the_tradeoff_between_reliability_and_data_rate)
- [4 Local decoding and testing of codes](https://en.wikipedia.org/wiki/Error_correction_code#Local_decoding_and_testing_of_codes)
- [5 Improving performance](https://en.wikipedia.org/wiki/Error_correction_code#Improving_performance)
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- [5\.1 Concatenation (combination)](https://en.wikipedia.org/wiki/Error_correction_code#Concatenation_\(combination\))
- [5\.2 Interleaving](https://en.wikipedia.org/wiki/Error_correction_code#Interleaving)
- [5\.2.1 Interleaving example](https://en.wikipedia.org/wiki/Error_correction_code#Interleaving_example)
- [5\.2.2 Disadvantages of interleaving](https://en.wikipedia.org/wiki/Error_correction_code#Disadvantages_of_interleaving)
- [6 Software for error-correcting codes](https://en.wikipedia.org/wiki/Error_correction_code#Software_for_error-correcting_codes)
- [7 List of error-correcting codes](https://en.wikipedia.org/wiki/Error_correction_code#List_of_error-correcting_codes)
- [8 See also](https://en.wikipedia.org/wiki/Error_correction_code#See_also)
- [9 References](https://en.wikipedia.org/wiki/Error_correction_code#References)
- [10 Further reading](https://en.wikipedia.org/wiki/Error_correction_code#Further_reading)
- [11 External links](https://en.wikipedia.org/wiki/Error_correction_code#External_links)
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From Wikipedia, the free encyclopedia
Scheme for controlling errors in data over noisy communication channels
"Interleaver" redirects here. For the fiber-optic device, see [optical interleaver](https://en.wikipedia.org/wiki/Optical_interleaver "Optical interleaver").
In [computing](https://en.wikipedia.org/wiki/Computing "Computing"), [telecommunication](https://en.wikipedia.org/wiki/Telecommunication "Telecommunication"), [information theory](https://en.wikipedia.org/wiki/Information_theory "Information theory"), and [coding theory](https://en.wikipedia.org/wiki/Coding_theory "Coding theory"), **forward error correction** (**FEC**) or **channel coding**[\[1\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-1) is a technique used for [controlling errors](https://en.wikipedia.org/wiki/Error_control "Error control") in [data transmission](https://en.wikipedia.org/wiki/Data_transmission "Data transmission") over unreliable or noisy [communication channels](https://en.wikipedia.org/wiki/Communication_channel "Communication channel").
The central idea is that the sender encodes the message in a [redundant](https://en.wikipedia.org/wiki/Redundancy_\(information_theory\) "Redundancy (information theory)") way, most often by using an **error correction code**, or **error correcting code** (**ECC**).[\[2\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-2)[\[3\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-Hamming-3) The redundancy allows the receiver not only to [detect errors](https://en.wikipedia.org/wiki/Error_detection "Error detection") that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a [reverse channel](https://en.wikipedia.org/wiki/Reverse_channel "Reverse channel") to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth.
The American mathematician [Richard Hamming](https://en.wikipedia.org/wiki/Richard_Hamming "Richard Hamming") pioneered this field in the 1940s and invented the first error-correcting code in 1950: the [Hamming (7,4) code](https://en.wikipedia.org/wiki/Hamming_\(7,4\)_code "Hamming (7,4) code").[\[3\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-Hamming-3)
FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in [multicast](https://en.wikipedia.org/wiki/Multicast "Multicast").
Long-latency connections also benefit; in the case of satellites orbiting distant planets, retransmission due to errors would create a delay of several hours. FEC is also widely used in [modems](https://en.wikipedia.org/wiki/Modem "Modem") and in [cellular networks](https://en.wikipedia.org/wiki/Cellular_network "Cellular network").
FEC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, FEC is an integral part of the initial [analog-to-digital conversion](https://en.wikipedia.org/wiki/Analog-to-digital_converter "Analog-to-digital converter") in the receiver. The [Viterbi decoder](https://en.wikipedia.org/wiki/Viterbi_decoder "Viterbi decoder") implements a [soft-decision algorithm](https://en.wikipedia.org/wiki/Soft-decision_decoder "Soft-decision decoder") to demodulate digital data from an analog signal corrupted by noise. Many FEC decoders can also generate a [bit-error rate](https://en.wikipedia.org/wiki/Bit-error_rate "Bit-error rate") (BER) signal which can be used as feedback to fine-tune the analog receiving electronics.
FEC information is added to [mass storage](https://en.wikipedia.org/wiki/Mass_storage "Mass storage") (magnetic, optical and solid state/flash based) devices to enable recovery of corrupted data, and is used as [ECC](https://en.wikipedia.org/wiki/ECC_memory "ECC memory") [computer memory](https://en.wikipedia.org/wiki/Computer_memory "Computer memory") on systems that require special provisions for reliability.
The maximum proportion of errors or missing bits that can be corrected is determined by the design of the ECC, so different forward error correcting codes are suitable for different conditions. In general, a stronger code induces more redundancy that needs to be transmitted using the available bandwidth, which reduces the effective bit-rate while improving the received effective [signal-to-noise ratio](https://en.wikipedia.org/wiki/Signal-to-noise_ratio "Signal-to-noise ratio"). The [noisy-channel coding theorem](https://en.wikipedia.org/wiki/Noisy-channel_coding_theorem "Noisy-channel coding theorem") of [Claude Shannon](https://en.wikipedia.org/wiki/Claude_Shannon "Claude Shannon") can be used to compute the maximum achievable communication bandwidth for a given maximum acceptable error probability. This establishes bounds on the theoretical maximum information transfer rate of a channel with some given base noise level. However, the proof is not constructive, and hence gives no insight of how to build a capacity achieving code. After years of research, some advanced FEC systems like [polar code](https://en.wikipedia.org/wiki/Polar_code_\(coding_theory\) "Polar code (coding theory)")[\[4\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-:0-4) come very close to the theoretical maximum given by the Shannon channel capacity under the hypothesis of an infinite length frame.
## Method
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=1 "Edit section: Method")\]
ECC is accomplished by adding [redundancy](https://en.wikipedia.org/wiki/Redundancy_\(information_theory\) "Redundancy (information theory)") to the transmitted information using an algorithm. A redundant bit may be a complicated function of many original information bits. The original information may or may not appear literally in the encoded output; codes that include the unmodified input in the output are **[systematic](https://en.wikipedia.org/wiki/Systematic_code "Systematic code")**, while those that do not are **non-systematic**.
A simplistic example of ECC is to transmit each data bit three times, which is known as a (3,1) [repetition code](https://en.wikipedia.org/wiki/Repetition_code "Repetition code"). Through a noisy channel, a receiver might see eight versions of the output; see the table below.
| Triplet received | Interpreted as |
|---|---|
| 000 | 0 (error-free) |
| 001 | 0 |
| 010 | 0 |
| 100 | 0 |
| 111 | 1 (error-free) |
| 110 | 1 |
| 101 | 1 |
| 011 | 1 |
This allows an error in any one of the three samples to be corrected by "majority vote" or "democratic voting". The correcting ability of this ECC is:
- up to one bit of triplet in error, or
- up to two bits of triplet omitted (cases not shown in table).
Though simple to implement and widely used, this [triple modular redundancy](https://en.wikipedia.org/wiki/Triple_modular_redundancy "Triple modular redundancy") is a relatively inefficient ECC. Better ECC codes typically examine the last several tens or even the last several hundreds of previously received bits to determine how to decode the current small handful of bits (typically in groups of two to eight bits).
### Simplified formalism
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=2 "Edit section: Simplified formalism")\]
Formally, an error-correcting code is given by its ([injective](https://en.wikipedia.org/wiki/Injective_function "Injective function")) encoding function f : Ī£ ā āŖ Ī ā {\\displaystyle f:\\Sigma ^{\*}\\hookrightarrow \\Delta ^{\*}}  which assigns to each word u ā Ī£ ā \= d e f ā n ā N Ī£ n {\\displaystyle u\\in \\Sigma ^{\*}\\;{\\stackrel {\\mathrm {def} }{=}}\\;\\textstyle \\bigcup \_{n\\in \\mathbb {N} }\\Sigma ^{n}}  of a finite [alphabet](https://en.wikipedia.org/wiki/Alphabet_\(formal_languages\) "Alphabet (formal languages)") Ī£ {\\displaystyle \\Sigma }  a unique word w ā Ī ā \= d e f ā n ā N Ī n {\\displaystyle w\\in \\Delta ^{\*}\\;{\\stackrel {\\mathrm {def} }{=}}\\;\\textstyle \\bigcup \_{n\\in \\mathbb {N} }\\Delta ^{n}}  (a concatenation of letters) from the [alphabet](https://en.wikipedia.org/wiki/Alphabet_\(formal_languages\) "Alphabet (formal languages)") Ī {\\displaystyle \\Delta } .
Most commonly, f {\\displaystyle f}  is a [homomorphism](https://en.wikipedia.org/wiki/Homomorphism "Homomorphism") in the sense that if x 1 ā x 2 {\\displaystyle x\_{1}\*x\_{2}}  is the concatenation of x 1 {\\displaystyle x\_{1}}  and x 2 {\\displaystyle x\_{2}} , then we have the following:f ( u 1 ā u 2 ) \= f ( u 1 ) ā f ( u 2 ) {\\displaystyle f(u\_{1}\*u\_{2})=f(u\_{1})\*f(u\_{2})} This implies that it is enough to define f ( Ļ ) {\\displaystyle f(\\sigma )}  for single-letter words Ļ {\\displaystyle \\sigma } . The [range](https://en.wikipedia.org/wiki/Range_of_a_function "Range of a function") I m ( f ) {\\displaystyle \\mathrm {Im} (f)}  of the function f {\\displaystyle f}  is the set of *code-words.* The capabilities of the code to detect and correct errors can then be understood from the *distance* d \= min w 1 , w 2 ā I m ( f ) ; w 1 ā w 2 d H ( w 1 , w 2 ) {\\displaystyle d=\\min \_{w\_{1},w\_{2}\\in \\mathrm {Im} (f);\\ w\_{1}\\neq w\_{2}}d\_{H}(w\_{1},w\_{2})}  of the code, which is the minimum [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance "Hamming distance") separating any two distinct code words. A code with distance d {\\displaystyle d}  can detect errors on k {\\displaystyle k}  bits as long as k \< d {\\displaystyle k\<d} , and among those detected errors, the code can correct l {\\displaystyle l} \-bit errors whenever 2 l \< d {\\displaystyle 2l\<d} .
### Averaging noise to reduce errors
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=3 "Edit section: Averaging noise to reduce errors")\]
ECC could be said to work by "averaging noise"; since each data bit affects many transmitted symbols, the corruption of some symbols by noise usually allows the original user data to be extracted from the other, uncorrupted received symbols that also depend on the same user data.
- Because of this "risk-pooling" effect, digital communication systems that use ECC tend to work well above a certain minimum [signal-to-noise ratio](https://en.wikipedia.org/wiki/Signal-to-noise_ratio "Signal-to-noise ratio") and not at all below it.
- This *all-or-nothing tendency* ā the [cliff effect](https://en.wikipedia.org/wiki/Cliff_effect "Cliff effect") ā becomes more pronounced as stronger codes are used that more closely approach the theoretical [Shannon limit](https://en.wikipedia.org/wiki/Shannon_limit "Shannon limit").
- Interleaving ECC coded data can reduce the all or nothing properties of transmitted ECC codes when the channel errors tend to occur in bursts. However, this method has limits; it is best used on narrowband data.
Most telecommunication systems use a fixed [channel code](https://en.wikipedia.org/wiki/Channel_code "Channel code") designed to tolerate the expected worst-case [bit error rate](https://en.wikipedia.org/wiki/Bit_error_rate "Bit error rate"), and then fail to work at all if the bit error rate is ever worse. However, some systems adapt to the given channel error conditions: some instances of [hybrid automatic repeat-request](https://en.wikipedia.org/wiki/Hybrid_automatic_repeat-request "Hybrid automatic repeat-request") use a fixed ECC method as long as the ECC can handle the error rate, then switch to [ARQ](https://en.wikipedia.org/wiki/Automatic_Repeat_Request "Automatic Repeat Request") when the error rate gets too high; [adaptive modulation and coding](https://en.wikipedia.org/wiki/Adaptive_modulation_and_coding "Adaptive modulation and coding") uses a variety of ECC rates, adding more error-correction bits per packet when there are higher error rates in the channel, or taking them out when they are not needed.
## Types
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=4 "Edit section: Types")\]
Main articles: [Block code](https://en.wikipedia.org/wiki/Block_code "Block code") and [Convolutional code](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code")
[](https://en.wikipedia.org/wiki/File:Block_code_error_correction.png)
A block code (specifically a [Hamming code](https://en.wikipedia.org/wiki/Hamming_code "Hamming code")) where redundant bits are added as a block to the end of the initial message
[](https://en.wikipedia.org/wiki/File:Convolutional_code_error_correction.png)
A continuous [convolutional code](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code") where redundant bits are added continuously into the structure of the code word
The two main categories of ECC codes are [block codes](https://en.wikipedia.org/wiki/Block_code "Block code") and [convolutional codes](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code").
- Block codes work on fixed-size blocks (packets) of bits or symbols of predetermined size. Practical block codes can generally be hard-decoded in [polynomial time](https://en.wikipedia.org/wiki/Polynomial_time "Polynomial time") to their block length.
- Convolutional codes work on bit or symbol streams of arbitrary length. They are most often soft decoded with the [Viterbi algorithm](https://en.wikipedia.org/wiki/Viterbi_algorithm "Viterbi algorithm"), though other algorithms are sometimes used. Viterbi decoding allows asymptotically optimal decoding efficiency with increasing constraint length of the convolutional code, but at the expense of [exponentially](https://en.wikipedia.org/wiki/Exponential_time "Exponential time") increasing complexity. A convolutional code that is terminated is also a 'block code' in that it encodes a block of input data, but the block size of a convolutional code is generally arbitrary, while block codes have a fixed size dictated by their algebraic characteristics. Types of termination for convolutional codes include "tail-biting" and "bit-flushing".
Classical block codes are usually decoded using **hard-decision** algorithms,[\[5\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-5) which means that for every input and output signal a hard decision is made whether it corresponds to a one or a zero bit. In contrast, convolutional codes are typically decoded using **soft-decision** algorithms like the Viterbi, MAP or [BCJR](https://en.wikipedia.org/wiki/BCJR_algorithm "BCJR algorithm") algorithms, which process (discretized) analog signals, and which allow for much higher error-correction performance than hard-decision decoding.
Nearly all classical block codes apply the algebraic properties of [finite fields](https://en.wikipedia.org/wiki/Finite_field "Finite field"). Hence classical block codes are often referred to as algebraic codes.
### Block codes
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=5 "Edit section: Block codes")\]
There are many types of block codes; [ReedāSolomon coding](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction "ReedāSolomon error correction") is noteworthy for its widespread use in [compact discs](https://en.wikipedia.org/wiki/Compact_disc "Compact disc"), [DVDs](https://en.wikipedia.org/wiki/DVD "DVD"), and [hard disk drives](https://en.wikipedia.org/wiki/Hard_disk_drive#Error_rates_and_handling "Hard disk drive"). Other examples of classical block codes include [Golay](https://en.wikipedia.org/wiki/Golay_code_\(disambiguation\) "Golay code (disambiguation)"), [BCH](https://en.wikipedia.org/wiki/BCH_code "BCH code"), [Multidimensional parity](https://en.wikipedia.org/wiki/Multidimensional_parity-check_code "Multidimensional parity-check code"), and [Hamming codes](https://en.wikipedia.org/wiki/Hamming_code "Hamming code").
Hamming ECC is commonly used to correct [ECC memory](https://en.wikipedia.org/wiki/ECC_memory "ECC memory") and early SLC [NAND flash](https://en.wikipedia.org/wiki/NAND_flash "NAND flash") memory errors.[\[6\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-6) This provides single-bit error correction and 2-bit error detection. Hamming codes are only suitable for more reliable [single-level cell](https://en.wikipedia.org/wiki/Single-level_cell "Single-level cell") (SLC) NAND. Denser [multi-level cell](https://en.wikipedia.org/wiki/Multi-level_cell "Multi-level cell") (MLC) NAND may use multi-bit correcting ECC such as [BCH](https://en.wikipedia.org/wiki/BCH_code "BCH code"), ReedāSolomon, or [LDPC](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code").[\[7\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-spansion-7)[\[8\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-8) NOR flash typically does not use any error correction.[\[7\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-spansion-7)
### Soft codes
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=6 "Edit section: Soft codes")\]
#### Low-density parity-check (LDPC)
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=7 "Edit section: Low-density parity-check (LDPC)")\]
Main article: [Low-density parity-check code](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code")
[Low-density parity-check](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code") (LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the [channel capacity](https://en.wikipedia.org/wiki/Shannon_limit "Shannon limit") (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length. Practical implementations rely heavily on decoding the constituent SPC codes in parallel.
LDPC codes were first introduced by [Robert G. Gallager](https://en.wikipedia.org/wiki/Robert_G._Gallager "Robert G. Gallager") in his PhD thesis in 1960, but due to the computational effort in implementing encoder and decoder and the introduction of [ReedāSolomon](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon "ReedāSolomon") codes, they were mostly ignored until the 1990s.
LDPC codes are now used in many recent high-speed communication standards, such as [DVB-S2](https://en.wikipedia.org/wiki/DVB-S2 "DVB-S2") (Digital Video Broadcasting ā Satellite ā Second Generation), [WiMAX](https://en.wikipedia.org/wiki/WiMAX "WiMAX") ([IEEE 802.16e](https://en.wikipedia.org/wiki/IEEE_802.16e "IEEE 802.16e") standard for microwave communications), High-Speed Wireless LAN ([IEEE 802.11n](https://en.wikipedia.org/wiki/IEEE_802.11n "IEEE 802.11n")),[\[9\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-9) [10GBase-T Ethernet](https://en.wikipedia.org/wiki/802.3an#10GBASE-T "802.3an") (802.3an) and [G.hn/G.9960](https://en.wikipedia.org/wiki/G.hn "G.hn") (ITU-T Standard for networking over power lines, phone lines and coaxial cable). Other LDPC codes are standardized for wireless communication standards within [3GPP](https://en.wikipedia.org/wiki/3GPP "3GPP") [MBMS](https://en.wikipedia.org/wiki/MBMS "MBMS") (see [fountain codes](https://en.wikipedia.org/wiki/Fountain_code#Fountain_codes_in_standards "Fountain code")).
#### Turbo code
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=8 "Edit section: Turbo code")\]
Main article: [Turbo code](https://en.wikipedia.org/wiki/Turbo_code "Turbo code")
[Turbo coding](https://en.wikipedia.org/wiki/Turbo_code "Turbo code") is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the [Shannon limit](https://en.wikipedia.org/wiki/Shannon_limit "Shannon limit"). Predating [LDPC codes](https://en.wikipedia.org/wiki/LDPC_codes "LDPC codes") in terms of practical application, they now provide similar performance.
One of the earliest commercial applications of turbo coding was the [CDMA2000 1x](https://en.wikipedia.org/wiki/CDMA2000 "CDMA2000") (TIA IS-2000) digital cellular technology developed by [Qualcomm](https://en.wikipedia.org/wiki/Qualcomm "Qualcomm") and sold by [Verizon Wireless](https://en.wikipedia.org/wiki/Verizon_Wireless "Verizon Wireless"), [Sprint](https://en.wikipedia.org/wiki/Sprint_Nextel "Sprint Nextel"), and other carriers. It is also used for the evolution of CDMA2000 1x specifically for Internet access, [1xEV-DO](https://en.wikipedia.org/wiki/Evolution-Data_Optimized "Evolution-Data Optimized") (TIA IS-856). Like 1x, EV-DO was developed by [Qualcomm](https://en.wikipedia.org/wiki/Qualcomm "Qualcomm"), and is sold by [Verizon Wireless](https://en.wikipedia.org/wiki/Verizon_Wireless "Verizon Wireless"), [Sprint](https://en.wikipedia.org/wiki/Sprint_Nextel "Sprint Nextel"), and other carriers (Verizon's marketing name for 1xEV-DO is *Broadband Access*, Sprint's consumer and business marketing names for 1xEV-DO are *Power Vision* and *Mobile Broadband*, respectively).
## Describing the performance of an ECC
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=9 "Edit section: Describing the performance of an ECC")\]
In contrast to classical block codes that often specify an error-detecting or error-correcting ability, many modern block codes such as [LDPC codes](https://en.wikipedia.org/wiki/LDPC_codes "LDPC codes") lack such guarantees. Instead, modern codes are evaluated in terms of their bit error rates.
Most forward error correction codes correct only bit-flips, but not bit-insertions or bit-deletions. In this setting, the [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance "Hamming distance") is the appropriate way to measure the [bit error rate](https://en.wikipedia.org/wiki/Bit_error_rate "Bit error rate"). A few forward error correction codes are designed to correct bit-insertions and bit-deletions, such as Marker Codes and Watermark Codes. The [Levenshtein distance](https://en.wikipedia.org/wiki/Levenshtein_distance "Levenshtein distance") is a more appropriate way to measure the bit error rate when using such codes. [\[10\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-10)
### Code-rate and the tradeoff between reliability and data rate
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=10 "Edit section: Code-rate and the tradeoff between reliability and data rate")\]
See also: [Bit rate Ā§ Information rate](https://en.wikipedia.org/wiki/Bit_rate#Information_rate "Bit rate")
The fundamental principle of ECC is to add redundant bits in order to help the decoder to find out the true message that was encoded by the transmitter. The code-rate of a given ECC system is defined as the ratio between the number of information bits and the total number of bits (i.e., information plus redundancy bits) in a given communication package. The code-rate is hence a real number. A low code-rate close to zero implies a strong code that uses many redundant bits to achieve a good performance, while a large code-rate close to 1 implies a weak code.
The redundant bits that protect the information have to be transferred using the same communication resources that they are trying to protect. This causes a fundamental tradeoff between reliability and data rate.[\[11\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-11) In one extreme, a strong code (with low code-rate) can induce an important increase in the receiver SNR (signal-to-noise-ratio) decreasing the bit error rate, at the cost of reducing the effective data rate. On the other extreme, not using any ECC (i.e., a code-rate equal to 1) uses the full channel for information transfer purposes, at the cost of leaving the bits without any additional protection.
One interesting question is the following: how efficient in terms of information transfer can an ECC be that has a negligible decoding error rate? This question was answered by Claude Shannon with his second theorem, which says that the channel capacity is the maximum bit rate achievable by any ECC whose error rate tends to zero:[\[12\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-shannon_paper-12) His proof relies on Gaussian random coding, which is not suitable to real-world applications. The upper bound given by Shannon's work inspired a long journey in designing ECCs that can come close to the ultimate performance boundary. Various codes today can attain almost the Shannon limit. However, capacity achieving ECCs are usually extremely complex to implement.
The most popular ECCs have a trade-off between performance and computational complexity. Usually, their parameters give a range of possible code rates, which can be optimized depending on the scenario. Usually, this optimization is done in order to achieve a low decoding error probability while minimizing the impact to the data rate. Another criterion for optimizing the code rate is to balance low error rate and retransmissions number in order to the energy cost of the communication.[\[13\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-13)
## Local decoding and testing of codes
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=11 "Edit section: Local decoding and testing of codes")\]
Main articles: [Locally decodable code](https://en.wikipedia.org/wiki/Locally_decodable_code "Locally decodable code") and [Locally testable code](https://en.wikipedia.org/wiki/Locally_testable_code "Locally testable code")
Sometimes it is only necessary to decode single bits of the message, or to check whether a given signal is a codeword, and do so without looking at the entire signal. This can make sense in a streaming setting, where codewords are too large to be classically decoded fast enough and where only a few bits of the message are of interest for now. Also such codes have become an important tool in [computational complexity theory](https://en.wikipedia.org/wiki/Computational_complexity_theory "Computational complexity theory"), e.g., for the design of [probabilistically checkable proofs](https://en.wikipedia.org/wiki/Probabilistically_checkable_proof "Probabilistically checkable proof").
[Locally decodable codes](https://en.wikipedia.org/wiki/Locally_decodable_code "Locally decodable code") are error-correcting codes for which single bits of the message can be probabilistically recovered by only looking at a small (say constant) number of positions of a codeword, even after the codeword has been corrupted at some constant fraction of positions. [Locally testable codes](https://en.wikipedia.org/wiki/Locally_testable_code "Locally testable code") are error-correcting codes for which it can be checked probabilistically whether a signal is close to a codeword by only looking at a small number of positions of the signal.
Not all locally decodable codes (LDCs) are locally testable codes (LTCs)[\[14\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-decNotTest-14) neither locally correctable codes (LCCs),[\[15\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-15) q-query LCCs are bounded exponentially[\[16\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-16) while LDCs can have [subexponential](https://en.wikipedia.org/wiki/Subexponential_time "Subexponential time") lengths.[\[17\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-17)
## Improving performance
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=12 "Edit section: Improving performance")\]
### Concatenation (combination)
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=13 "Edit section: Concatenation (combination)")\]
Main article: [Concatenated error correction codes](https://en.wikipedia.org/wiki/Concatenated_error_correction_codes "Concatenated error correction codes")
Classical (algebraic) block codes and convolutional codes are frequently combined in **concatenated** coding schemes in which a short constraint-length Viterbi-decoded convolutional code does most of the work and a block code (usually ReedāSolomon) with larger symbol size and block length "mops up" any errors made by the convolutional decoder. Single pass decoding with this family of error correction codes can yield very low error rates, but for long range transmission conditions (like deep space) iterative decoding is recommended.
Concatenated codes have been standard practice in satellite and deep space communications since [Voyager 2](https://en.wikipedia.org/wiki/Voyager_program "Voyager program") first used the technique in its 1986 encounter with [Uranus](https://en.wikipedia.org/wiki/Uranus "Uranus"). The [Galileo](https://en.wikipedia.org/wiki/Galileo_\(spacecraft\) "Galileo (spacecraft)") craft used iterative concatenated codes to compensate for the very high error rate conditions caused by having a failed antenna.
### Interleaving
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=14 "Edit section: Interleaving")\]
"Interleaver" redirects here. For the fiber-optic device, see [optical interleaver](https://en.wikipedia.org/wiki/Optical_interleaver "Optical interleaver").
[](https://en.wikipedia.org/wiki/File:Interleaving1.png)
A short illustration of the interleaving idea
Interleaving is frequently used in digital communication and storage systems to improve the performance of forward error correcting codes. Many [communication channels](https://en.wikipedia.org/wiki/Communication_channel "Communication channel") are not memoryless: errors typically occur in [bursts](https://en.wikipedia.org/wiki/Burst_error "Burst error") rather than independently. If the number of errors within a [code word](https://en.wikipedia.org/wiki/Code_word_\(communication\) "Code word (communication)") exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving alleviates this problem by shuffling source symbols across several code words, thereby creating a more [uniform distribution](https://en.wikipedia.org/wiki/Uniform_distribution_\(continuous\) "Uniform distribution (continuous)") of errors.[\[18\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-turbo-principles-18) Therefore, interleaving is widely used for [burst error-correction](https://en.wikipedia.org/wiki/Burst_error-correcting_code "Burst error-correcting code").
The analysis of modern iterated codes, like [turbo codes](https://en.wikipedia.org/wiki/Turbo_code "Turbo code") and [LDPC codes](https://en.wikipedia.org/wiki/LDPC_code "LDPC code"), typically assumes an independent distribution of errors.[\[19\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-19) Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word.[\[20\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-20)
For turbo codes, an interleaver is an integral component and its proper design is crucial for good performance.[\[18\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-turbo-principles-18)[\[21\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-21) The iterative decoding algorithm works best when there are not short cycles in the [factor graph](https://en.wikipedia.org/wiki/Factor_graph "Factor graph") that represents the decoder; the interleaver is chosen to avoid short cycles.
Interleaver designs include:
- rectangular (or uniform) interleavers (similar to the method using skip factors described above)
- convolutional interleavers
- random interleavers (where the interleaver is a known random permutation)
- S-random interleaver (where the interleaver is a known random permutation with the constraint that no input symbols within distance S appear within a distance of S in the output).[\[22\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-22)
- a contention-free quadratic [permutation polynomial](https://en.wikipedia.org/wiki/Permutation_polynomial "Permutation polynomial") (QPP).[\[23\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-Takeshita1-23) An example of use is in the [3GPP Long Term Evolution](https://en.wikipedia.org/wiki/3GPP_Long_Term_Evolution "3GPP Long Term Evolution") mobile telecommunication standard.[\[24\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-24)
In multi-[carrier](https://en.wikipedia.org/wiki/Carrier_signal "Carrier signal") communication systems, interleaving across carriers may be employed to provide frequency [diversity](https://en.wikipedia.org/wiki/Diversity_scheme "Diversity scheme"), e.g., to mitigate [frequency-selective fading](https://en.wikipedia.org/wiki/Frequency-selective_fading "Frequency-selective fading") or narrowband interference.[\[25\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-25)
#### Interleaving example
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=15 "Edit section: Interleaving example")\]
**Transmission without interleaving**:
```
Error-free message: aaaabbbbccccddddeeeeffffgggg
Transmission with a burst error: aaaabbbbccc____deeeeffffgggg
```
Here, each group of the same letter represents a 4-bit one-bit error-correcting codeword. The codeword cccc is altered in one bit and can be corrected, but the codeword dddd is altered in three bits, so either it cannot be decoded at all or it might be [decoded incorrectly](https://en.wikipedia.org/wiki/Falsing "Falsing").
**With interleaving**:
```
Error-free code words: aaaabbbbccccddddeeeeffffgggg
Interleaved: abcdefgabcdefgabcdefgabcdefg
Transmission with a burst error: abcdefgabcd____bcdefgabcdefg
Received code words after deinterleaving: aa_abbbbccccdddde_eef_ffg_gg
```
In each of the codewords "aaaa", "eeee", "ffff", and "gggg", only one bit is altered, so one-bit error-correcting code will decode everything correctly.
**Transmission without interleaving**:
```
Original transmitted sentence: ThisIsAnExampleOfInterleaving
Received sentence with a burst error: ThisIs______pleOfInterleaving
```
The term "AnExample" ends up mostly unintelligible and difficult to correct.
**With interleaving**:
```
Transmitted sentence: ThisIsAnExampleOfInterleaving...
Error-free transmission: TIEpfeaghsxlIrv.iAaenli.snmOten.
Received sentence with a burst error: TIEpfe______Irv.iAaenli.snmOten.
Received sentence after deinterleaving: T_isI_AnE_amp_eOfInterle_vin_...
```
No word is completely lost and the missing letters can be recovered with minimal guesswork.
#### Disadvantages of interleaving
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=16 "Edit section: Disadvantages of interleaving")\]
Use of interleaving techniques increases total delay. This is because the entire interleaved block must be received before the packets can be decoded.[\[26\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-26) Also interleavers hide the structure of errors; without an interleaver, more advanced decoding algorithms can take advantage of the error structure and achieve more reliable communication than a simpler decoder combined with an interleaver\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]. An example of such an algorithm is based on [neural network](https://en.wikipedia.org/wiki/Neural_network "Neural network")[\[27\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-27) structures.
## Software for error-correcting codes
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=17 "Edit section: Software for error-correcting codes")\]
Simulating the behaviour of error-correcting codes (ECCs) in software is a common practice to design, validate and improve ECCs. The upcoming wireless 5G standard raises a new range of applications for the software ECCs: the [Cloud Radio Access Networks (C-RAN)](https://en.wikipedia.org/wiki/C-RAN "C-RAN") in a [Software-defined radio (SDR)](https://en.wikipedia.org/wiki/Software-defined_radio "Software-defined radio") context. The idea is to directly use software ECCs in the communications. For instance in the 5G, the software ECCs could be located in the cloud and the antennas connected to this computing resources: improving this way the flexibility of the communication network and eventually increasing the energy efficiency of the system.
In this context, there are various available Open-source software listed below (non exhaustive).
- [AFF3CT](https://aff3ct.github.io/)(A Fast Forward Error Correction Toolbox): a full communication chain in C++ (many supported codes like Turbo, LDPC, Polar codes, etc.), very fast and specialized on channel coding (can be used as a program for simulations or as a library for the SDR).
- [IT++](https://en.wikipedia.org/wiki/IT%2B%2B "IT++"): a C++ library of classes and functions for linear algebra, numerical optimization, signal processing, communications, and statistics.
- [OpenAir](https://gitlab.eurecom.fr/oai/openair-cn): implementation (in C) of the 3GPP specifications concerning the Evolved Packet Core Networks.
## List of error-correcting codes
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=18 "Edit section: List of error-correcting codes")\]
| Code | Distance | Detectable errors (bits) | Correctable errors (bits) |
|---|---|---|---|
| Parity (guess needed on error) | 2 | 1 | 0 |
| [Triple modular redundancy](https://en.wikipedia.org/wiki/Triple_modular_redundancy "Triple modular redundancy") | 3 | 2 | 1 |
| Perfect Hamming such as [Hamming(7,4)](https://en.wikipedia.org/wiki/Hamming\(7,4\) "Hamming(7,4)") | 3 | 2 | 1 |
| [SECDED](https://en.wikipedia.org/wiki/SECDED "SECDED"): extended Hamming such as (39,32), (72,64) | 4 | 3 | 1 |
| DECTED: [Nordstrom-Robinson code](https://en.wikipedia.org/wiki/Nordstrom-Robinson_code "Nordstrom-Robinson code") | 6 | 5 | 2 |
| Perfect [binary Golay code](https://en.wikipedia.org/wiki/Binary_Golay_code "Binary Golay code") | 7 | 6 | 3 |
| TECFED: Extended [binary Golay code](https://en.wikipedia.org/wiki/Binary_Golay_code "Binary Golay code") | 8 | 7 | 3 |
In the table above, for an error-correcting code of minimal Hamming distance d {\\displaystyle d} , the maximum number of errors that the code can detect is given by d ā 1 {\\displaystyle d-1}  while the maximal number of errors that it can correct is given by ā d ā 1 2 ā {\\displaystyle \\left\\lfloor {\\frac {d-1}{2}}\\right\\rfloor } .
- [AN codes](https://en.wikipedia.org/wiki/AN_codes "AN codes")
- [Algebraic geometry code](https://en.wikipedia.org/wiki/Algebraic_geometry_code "Algebraic geometry code")
- [BCH code](https://en.wikipedia.org/wiki/BCH_code "BCH code"), which can be designed to correct any arbitrary number of errors per code block.
- [Barker code](https://en.wikipedia.org/wiki/Barker_code "Barker code") used for radar, telemetry, ultra sound, Wifi, DSSS mobile phone networks, GPS etc.
- [Berger code](https://en.wikipedia.org/wiki/Berger_code "Berger code")
- [Constant-weight code](https://en.wikipedia.org/wiki/Constant-weight_code "Constant-weight code")
- [Convolutional code](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code")
- [Expander codes](https://en.wikipedia.org/wiki/Expander_code "Expander code")
- [Group codes](https://en.wikipedia.org/wiki/Group_code "Group code")
- [Golay codes](https://en.wikipedia.org/wiki/Golay_code_\(disambiguation\) "Golay code (disambiguation)"), of which the [Binary Golay code](https://en.wikipedia.org/wiki/Binary_Golay_code "Binary Golay code") is of practical interest
- [Goppa code](https://en.wikipedia.org/wiki/Binary_Goppa_code "Binary Goppa code"), used in the [McEliece cryptosystem](https://en.wikipedia.org/wiki/McEliece_cryptosystem "McEliece cryptosystem")
- [Hadamard code](https://en.wikipedia.org/wiki/Hadamard_code "Hadamard code")
- [Hagelbarger code](https://en.wikipedia.org/wiki/Hagelbarger_code "Hagelbarger code")
- [Hamming code](https://en.wikipedia.org/wiki/Hamming_code "Hamming code")
- [Latin square based code](https://en.wikipedia.org/wiki/Latin_square#Error_correcting_codes "Latin square") for non-white noise (prevalent for example in broadband over powerlines)
- [Lexicographic code](https://en.wikipedia.org/wiki/Lexicographic_code "Lexicographic code")
- [Linear Network Coding](https://en.wikipedia.org/wiki/Linear_network_coding "Linear network coding"), a type of erasure correcting code across networks instead of point-to-point links
- [Long code](https://en.wikipedia.org/wiki/Long_code_\(mathematics\) "Long code (mathematics)")
- [Low-density parity-check code](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code"), also known as [Gallager code](https://en.wikipedia.org/wiki/Gallager_code "Gallager code"), as the archetype for [sparse graph codes](https://en.wikipedia.org/wiki/Sparse_graph_code "Sparse graph code")
- [LT code](https://en.wikipedia.org/wiki/LT_code "LT code"), which is a near-optimal [rateless erasure correcting code (Fountain code)](https://en.wikipedia.org/wiki/Fountain_code "Fountain code")
- [m of n codes](https://en.wikipedia.org/wiki/M_of_n_codes "M of n codes")
- [Nordstrom-Robinson code](https://en.wikipedia.org/wiki/Nordstrom-Robinson_code "Nordstrom-Robinson code"), used in Geometry and Group Theory[\[28\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-28)
- [Online code](https://en.wikipedia.org/wiki/Online_code "Online code"), a near-optimal [rateless erasure correcting code](https://en.wikipedia.org/wiki/Fountain_code "Fountain code")
- [Polar code (coding theory)](https://en.wikipedia.org/wiki/Polar_code_\(coding_theory\) "Polar code (coding theory)")
- [Raptor code](https://en.wikipedia.org/wiki/Raptor_code "Raptor code"), a near-optimal [rateless erasure correcting code](https://en.wikipedia.org/wiki/Fountain_code "Fountain code")
- [ReedāSolomon error correction](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction "ReedāSolomon error correction")
- [ReedāMuller code](https://en.wikipedia.org/wiki/Reed%E2%80%93Muller_code "ReedāMuller code")
- [Repeat-accumulate code](https://en.wikipedia.org/wiki/Repeat-accumulate_code "Repeat-accumulate code")
- [Repetition codes](https://en.wikipedia.org/wiki/Repetition_code "Repetition code"), such as [Triple modular redundancy](https://en.wikipedia.org/wiki/Triple_modular_redundancy "Triple modular redundancy")
- Spinal code, a rateless, nonlinear code based on pseudo-random hash functions[\[29\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-29)
- [Tornado code](https://en.wikipedia.org/wiki/Tornado_code "Tornado code"), a near-optimal [erasure correcting code](https://en.wikipedia.org/wiki/Erasure_code "Erasure code"), and the precursor to [Fountain codes](https://en.wikipedia.org/wiki/Fountain_code "Fountain code")
- [Turbo code](https://en.wikipedia.org/wiki/Turbo_code "Turbo code")
- [WalshāHadamard code](https://en.wikipedia.org/wiki/Walsh%E2%80%93Hadamard_code "WalshāHadamard code")
- [Cyclic redundancy checks](https://en.wikipedia.org/wiki/Cyclic_redundancy_check "Cyclic redundancy check") (CRCs) can correct 1-bit errors for messages at most
2
n
ā
1
ā
1
{\\displaystyle 2^{n-1}-1}
bits long for optimal generator polynomials of degree
n
{\\displaystyle n}
, see [Mathematics of cyclic redundancy checks Ā§ Bitfilters](https://en.wikipedia.org/wiki/Mathematics_of_cyclic_redundancy_checks#Bitfilters "Mathematics of cyclic redundancy checks")
- [Locally Recoverable Codes](https://en.wikipedia.org/wiki/Locally_Recoverable_Codes "Locally Recoverable Codes")
- [Message authentication code](https://en.wikipedia.org/wiki/Message_authentication_code "Message authentication code")
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=19 "Edit section: See also")\]
- [Burst error-correcting code](https://en.wikipedia.org/wiki/Burst_error-correcting_code "Burst error-correcting code")
- [Code rate](https://en.wikipedia.org/wiki/Code_rate "Code rate")
- [Erasure codes](https://en.wikipedia.org/wiki/Erasure_code "Erasure code")
- [Error detection and correction](https://en.wikipedia.org/wiki/Error_detection_and_correction "Error detection and correction")
- [Error-correcting codes with feedback](https://en.wikipedia.org/wiki/Error-correcting_codes_with_feedback "Error-correcting codes with feedback")
- [Linear code](https://en.wikipedia.org/wiki/Linear_code "Linear code")
- [Quantum error correction](https://en.wikipedia.org/wiki/Quantum_error_correction "Quantum error correction")
- [Soft-decision decoder](https://en.wikipedia.org/wiki/Soft-decision_decoder "Soft-decision decoder")
## References
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=20 "Edit section: References")\]
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Charles Wang; Dean Sklar; Diana Johnson (Winter 2001ā2002). ["Forward Error-Correction Coding"](https://web.archive.org/web/20120314085127/http://www.aero.org/publications/crosslink/winter2002/04.html). *Crosslink*. **3** (1). The Aerospace Corporation. Archived from [the original](http://www.aero.org/publications/crosslink/winter2002/04.html) on 14 March 2012. Retrieved 5 March 2006.
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Glover, Neal; Dudley, Trent (1990). *Practical Error Correction Design For Engineers* (Revision 1.1, 2nd ed.). CO, USA: [Cirrus Logic](https://en.wikipedia.org/wiki/Cirrus_Logic "Cirrus Logic"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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[Hamming, Richard Wesley](https://en.wikipedia.org/wiki/Richard_Wesley_Hamming "Richard Wesley Hamming") (April 1950). "Error Detecting and Error Correcting Codes". *[Bell System Technical Journal](https://en.wikipedia.org/wiki/Bell_System_Technical_Journal "Bell System Technical Journal")*. **29** (2). USA: [AT\&T](https://en.wikipedia.org/wiki/AT%26T "AT&T"): 147ā160\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1950BSTJ...29..147H](https://ui.adsabs.harvard.edu/abs/1950BSTJ...29..147H). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/j.1538-7305.1950.tb00463.x](https://doi.org/10.1002%2Fj.1538-7305.1950.tb00463.x). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10945/46756](https://hdl.handle.net/10945%2F46756). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [61141773](https://api.semanticscholar.org/CorpusID:61141773).
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Maunder, Robert (2016). ["Overview of Channel Coding"](https://www.accelercomm.com/overview-channel-coding).
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Baldi, M.; Chiaraluce, F. (2008). ["A Simple Scheme for Belief Propagation Decoding of BCH and RS Codes in Multimedia Transmissions"](https://doi.org/10.1155%2F2008%2F957846). *[International Journal of Digital Multimedia Broadcasting](https://en.wikipedia.org/wiki/International_Journal_of_Digital_Multimedia_Broadcasting "International Journal of Digital Multimedia Broadcasting")*. **2008**: 1ā12\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1155/2008/957846](https://doi.org/10.1155%2F2008%2F957846).
6. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-6)** ["Hamming codes for NAND flash memory devices"](http://www.eetasia.com/ART_8800575062_499486_AN_7549c493.HTM) [Archived](https://web.archive.org/web/20160821122453/http://www.eetasia.com/ART_8800575062_499486_AN_7549c493.HTM) 21 August 2016 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine"). EE Times-Asia. Apparently based on ["Micron Technical Note TN-29-08: Hamming Codes for NAND Flash Memory Devices"](http://www.micron.com/~/media/Documents/Products/Technical%20Note/NAND%20Flash/tn2908_NAND_hamming_ECC_code.pdf) [Archived](https://web.archive.org/web/20170829073235/http://www.micron.com/~/media/Documents/Products/Technical%20Note/NAND%20Flash/tn2908_NAND_hamming_ECC_code.pdf) 29 August 2017 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine"). 2005. Both say: "The Hamming algorithm is an industry-accepted method for error detection and correction in many SLC NAND flash-based applications."
7. ^ [***a***](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-spansion_7-0) [***b***](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-spansion_7-1)
["What Types of ECC Should Be Used on Flash Memory?"](https://web.archive.org/web/20160614220343/http://www.spansion.com/Support/Application%20Notes/Types_of_ECC_Used_on_Flash_AN.pdf) (PDF). Spansion. 2011. Archived from [the original](http://www.spansion.com/Support/Application%20Notes/Types_of_ECC_Used_on_Flash_AN.pdf) (Application note) on 14 June 2016. "Both ReedāSolomon algorithm and BCH algorithm are common ECC choices for MLC NAND flash. ... Hamming based block codes are the most commonly used ECC for SLC.... both ReedāSolomon and BCH are able to handle multiple errors and are widely used on MLC flash."
8. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-8)**
Jim Cooke (August 2007). ["The Inconvenient Truths of NAND Flash Memory"](https://cushychicken.github.io/assets/cooke_inconvenient_truths.pdf) (PDF). p. 28. "For SLC, a code with a correction threshold of 1 is sufficient. t=4 required ... for MLC."
9. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-9)** IEEE Standard, section 20.3.11.6 ["802.11n-2009"](http://standards.ieee.org/getieee802/download/802.11n-2009.pdf) [Archived](https://web.archive.org/web/20130203104520/http://standards.ieee.org/getieee802/download/802.11n-2009.pdf) 3 February 2013 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine"), IEEE, 29 October 2009, accessed 21 March 2011.
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Shah, Gaurav; Molina, Andres; Blaze, Matt (2006). ["Keyboards and covert channels"](https://www.usenix.org/legacy/event/sec06/tech/full_papers/shah/shah_html/jbug-Usenix06.html). *USENIX*. Retrieved 20 December 2018.
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Tse, David; Viswanath, Pramod (2005), *Fundamentals of Wireless Communication*, [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"), UK
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Shannon, C. E. (1948). ["A mathematical theory of communication"](http://www.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf) (PDF). *[Bell System Technical Journal](https://en.wikipedia.org/wiki/Bell_System_Technical_Journal "Bell System Technical Journal")*. **27** (3ā4\): 379ā423 & 623ā656. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1948BSTJ...27..379S](https://ui.adsabs.harvard.edu/abs/1948BSTJ...27..379S). [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)"):[11858/00-001M-0000-002C-4314-2](https://hdl.handle.net/11858%2F00-001M-0000-002C-4314-2).
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Rosas, F.; Brante, G.; Souza, R. D.; Oberli, C. (2014). "Optimizing the code rate for achieving energy-efficient wireless communications". *Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC)*. pp. 775ā780\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/WCNC.2014.6952166](https://doi.org/10.1109%2FWCNC.2014.6952166). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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.
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[Kaufman, Tali](https://en.wikipedia.org/wiki/Tali_Kaufman "Tali Kaufman"); Viderman, Michael. ["Locally Testable vs. Locally Decodable Codes"](http://eccc.hpi-web.de/report/2010/130/revision/1/download/).
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Brubaker, Ben (9 January 2024). ["'Magical' Error Correction Scheme Proved Inherently Inefficient"](https://www.quantamagazine.org/magical-error-correction-scheme-proved-inherently-inefficient-20240109/). *Quanta Magazine*. Retrieved 9 January 2024.
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Kothari, Pravesh; Manohar, Peter (11 June 2024). ["An Exponential Lower Bound for Linear 3-Query Locally Correctable Codes"](https://dl.acm.org/doi/epdf/10.1145/3618260.3649640). *Symposium on Theory of Computing*. **56**: 776ā787\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1145/3618260.3649640](https://doi.org/10.1145%2F3618260.3649640). Retrieved 24 March 2026.
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Vucetic, B.; Yuan, J. (2000). *Turbo codes: principles and applications*. [Springer Verlag](https://en.wikipedia.org/wiki/Springer_Verlag "Springer Verlag"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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Dolinar, S.; Divsalar, D. (15 August 1995). "Weight Distributions for Turbo Codes Using Random and Nonrandom Permutations". *TDA Progress Report*. **122**: 42ā122\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1995TDAPR.122...56D](https://ui.adsabs.harvard.edu/abs/1995TDAPR.122...56D). [CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_\(identifier\) "CiteSeerX (identifier)") [10\.1.1.105.6640](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.105.6640).
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Krastanov, Stefan; Jiang, Liang (8 September 2017). ["Deep Neural Network Probabilistic Decoder for Stabilizer Codes"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5591216). *Scientific Reports*. **7** (1): 11003. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1705\.09334](https://arxiv.org/abs/1705.09334). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2017NatSR...711003K](https://ui.adsabs.harvard.edu/abs/2017NatSR...711003K). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/s41598-017-11266-1](https://doi.org/10.1038%2Fs41598-017-11266-1). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [5591216](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5591216). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [28887480](https://pubmed.ncbi.nlm.nih.gov/28887480).
28. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-28)**
Nordstrom, A.W.; Robinson, J.P. (1967), "An optimum nonlinear code", *Information and Control*, **11** (5ā6\): 613ā616, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/S0019-9958(67)90835-2](https://doi.org/10.1016%2FS0019-9958%2867%2990835-2)
29. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-29)**
Perry, Jonathan; Balakrishnan, Hari; Shah, Devavrat (2011). ["Rateless Spinal Codes"](http://doi.acm.org/10.1145/2070562.2070568). *Proceedings of the 10th ACM Workshop on Hot Topics in Networks*. pp. 1ā6\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1145/2070562.2070568](https://doi.org/10.1145%2F2070562.2070568). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[1721\.1/79676](https://hdl.handle.net/1721.1%2F79676). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781450310598](https://en.wikipedia.org/wiki/Special:BookSources/9781450310598 "Special:BookSources/9781450310598")
.
## Further reading
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=21 "Edit section: Further reading")\]
- [MacWilliams, Florence Jessiem](https://en.wikipedia.org/wiki/Florence_Jessiem_MacWilliams "Florence Jessiem MacWilliams"); [Sloane, Neil James Alexander](https://en.wikipedia.org/wiki/Neil_James_Alexander_Sloane "Neil James Alexander Sloane") (2007) \[1977\]. Written at AT\&T Shannon Labs, Florham Park, New Jersey, USA. *The Theory of Error-Correcting Codes*. North-Holland Mathematical Library. Vol. 16 (digital print of 12th impression, 1st ed.). Amsterdam / London / New York / Tokyo: [North-Holland](https://en.wikipedia.org/wiki/North-Holland_Publishing_Company "North-Holland Publishing Company") / [Elsevier BV](https://en.wikipedia.org/wiki/Elsevier_BV "Elsevier BV"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-444-85193-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-444-85193-2 "Special:BookSources/978-0-444-85193-2")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [76-41296](https://lccn.loc.gov/76-41296).
(xxii+762+6 pages)
- Clark, Jr., George C.; Cain, J. Bibb (1981). *Error-Correction Coding for Digital Communications*. New York, USA: [Plenum Press](https://en.wikipedia.org/wiki/Plenum_Press "Plenum Press"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-306-40615-2](https://en.wikipedia.org/wiki/Special:BookSources/0-306-40615-2 "Special:BookSources/0-306-40615-2")
.
- Arazi, Benjamin (1987). Swetman, Herb (ed.). *A Commonsense Approach to the Theory of Error Correcting Codes*. [MIT Press](https://en.wikipedia.org/wiki/MIT_Press "MIT Press") Series in Computer Systems. Vol. 10 (1 ed.). Cambridge, Massachusetts, USA / London, UK: [Massachusetts Institute of Technology](https://en.wikipedia.org/wiki/Massachusetts_Institute_of_Technology "Massachusetts Institute of Technology"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-262-01098-4](https://en.wikipedia.org/wiki/Special:BookSources/0-262-01098-4 "Special:BookSources/0-262-01098-4")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [87-21889](https://lccn.loc.gov/87-21889).
(x+2+208+4 pages)
- Wicker, Stephen B. (1995). *Error Control Systems for Digital Communication and Storage*. Englewood Cliffs, New Jersey, USA: [Prentice-Hall](https://en.wikipedia.org/wiki/Prentice-Hall "Prentice-Hall"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-13-200809-2](https://en.wikipedia.org/wiki/Special:BookSources/0-13-200809-2 "Special:BookSources/0-13-200809-2")
.
- Wilson, Stephen G. (1996). *Digital Modulation and Coding*. Englewood Cliffs, New Jersey, USA: [Prentice-Hall](https://en.wikipedia.org/wiki/Prentice-Hall "Prentice-Hall"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-13-210071-1](https://en.wikipedia.org/wiki/Special:BookSources/0-13-210071-1 "Special:BookSources/0-13-210071-1")
.
- ["Error Correction Code in Single Level Cell NAND Flash memories"](https://web.archive.org/web/20070519161253/http://www.st.com/stonline/products/literature/an/10123.htm) 2007-02-16
- ["Error Correction Code in NAND Flash memories"](https://web.archive.org/web/20160304050018/http://www.eetasia.com/ARTICLES/2004NOV/A/2004NOV29_MEM_AN10.PDF?SOURCES=DOWNLOAD) 2004-11-29
- [Observations on Errors, Corrections, & Trust of Dependent Systems](http://perspectives.mvdirona.com/2012/02/observations-on-errors-corrections-trust-of-dependent-systems/), by James Hamilton, 2012-02-26
- *[Sphere Packings, Lattices and Groups](https://en.wikipedia.org/wiki/Sphere_Packings,_Lattices_and_Groups "Sphere Packings, Lattices and Groups"),* by J. H. Conway, Neil James Alexander Sloane, [Springer Science & Business Media](https://en.wikipedia.org/wiki/Springer_Science_%26_Business_Media "Springer Science & Business Media"), 2013-03-09 ā Mathematics ā 682 pages.
## External links
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=22 "Edit section: External links")\]
- Morelos-Zaragoza, Robert (2004). ["The Correcting Codes (ECC) Page"](http://www.eccpage.com/). Retrieved 5 March 2006.
- [error correction zoo](https://errorcorrectionzoo.org/). Database of error correcting codes.
- [lpdec: library for LP decoding and related things (Python)](https://github.com/supermihi/lpdec)
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Error correction code
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| Readable Markdown | "Interleaver" redirects here. For the fiber-optic device, see [optical interleaver](https://en.wikipedia.org/wiki/Optical_interleaver "Optical interleaver").
In [computing](https://en.wikipedia.org/wiki/Computing "Computing"), [telecommunication](https://en.wikipedia.org/wiki/Telecommunication "Telecommunication"), [information theory](https://en.wikipedia.org/wiki/Information_theory "Information theory"), and [coding theory](https://en.wikipedia.org/wiki/Coding_theory "Coding theory"), **forward error correction** (**FEC**) or **channel coding**[\[1\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-1) is a technique used for [controlling errors](https://en.wikipedia.org/wiki/Error_control "Error control") in [data transmission](https://en.wikipedia.org/wiki/Data_transmission "Data transmission") over unreliable or noisy [communication channels](https://en.wikipedia.org/wiki/Communication_channel "Communication channel").
The central idea is that the sender encodes the message in a [redundant](https://en.wikipedia.org/wiki/Redundancy_\(information_theory\) "Redundancy (information theory)") way, most often by using an **error correction code**, or **error correcting code** (**ECC**).[\[2\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-2)[\[3\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-Hamming-3) The redundancy allows the receiver not only to [detect errors](https://en.wikipedia.org/wiki/Error_detection "Error detection") that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a [reverse channel](https://en.wikipedia.org/wiki/Reverse_channel "Reverse channel") to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth.
The American mathematician [Richard Hamming](https://en.wikipedia.org/wiki/Richard_Hamming "Richard Hamming") pioneered this field in the 1940s and invented the first error-correcting code in 1950: the [Hamming (7,4) code](https://en.wikipedia.org/wiki/Hamming_\(7,4\)_code "Hamming (7,4) code").[\[3\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-Hamming-3)
FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in [multicast](https://en.wikipedia.org/wiki/Multicast "Multicast").
Long-latency connections also benefit; in the case of satellites orbiting distant planets, retransmission due to errors would create a delay of several hours. FEC is also widely used in [modems](https://en.wikipedia.org/wiki/Modem "Modem") and in [cellular networks](https://en.wikipedia.org/wiki/Cellular_network "Cellular network").
FEC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, FEC is an integral part of the initial [analog-to-digital conversion](https://en.wikipedia.org/wiki/Analog-to-digital_converter "Analog-to-digital converter") in the receiver. The [Viterbi decoder](https://en.wikipedia.org/wiki/Viterbi_decoder "Viterbi decoder") implements a [soft-decision algorithm](https://en.wikipedia.org/wiki/Soft-decision_decoder "Soft-decision decoder") to demodulate digital data from an analog signal corrupted by noise. Many FEC decoders can also generate a [bit-error rate](https://en.wikipedia.org/wiki/Bit-error_rate "Bit-error rate") (BER) signal which can be used as feedback to fine-tune the analog receiving electronics.
FEC information is added to [mass storage](https://en.wikipedia.org/wiki/Mass_storage "Mass storage") (magnetic, optical and solid state/flash based) devices to enable recovery of corrupted data, and is used as [ECC](https://en.wikipedia.org/wiki/ECC_memory "ECC memory") [computer memory](https://en.wikipedia.org/wiki/Computer_memory "Computer memory") on systems that require special provisions for reliability.
The maximum proportion of errors or missing bits that can be corrected is determined by the design of the ECC, so different forward error correcting codes are suitable for different conditions. In general, a stronger code induces more redundancy that needs to be transmitted using the available bandwidth, which reduces the effective bit-rate while improving the received effective [signal-to-noise ratio](https://en.wikipedia.org/wiki/Signal-to-noise_ratio "Signal-to-noise ratio"). The [noisy-channel coding theorem](https://en.wikipedia.org/wiki/Noisy-channel_coding_theorem "Noisy-channel coding theorem") of [Claude Shannon](https://en.wikipedia.org/wiki/Claude_Shannon "Claude Shannon") can be used to compute the maximum achievable communication bandwidth for a given maximum acceptable error probability. This establishes bounds on the theoretical maximum information transfer rate of a channel with some given base noise level. However, the proof is not constructive, and hence gives no insight of how to build a capacity achieving code. After years of research, some advanced FEC systems like [polar code](https://en.wikipedia.org/wiki/Polar_code_\(coding_theory\) "Polar code (coding theory)")[\[4\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-:0-4) come very close to the theoretical maximum given by the Shannon channel capacity under the hypothesis of an infinite length frame.
ECC is accomplished by adding [redundancy](https://en.wikipedia.org/wiki/Redundancy_\(information_theory\) "Redundancy (information theory)") to the transmitted information using an algorithm. A redundant bit may be a complicated function of many original information bits. The original information may or may not appear literally in the encoded output; codes that include the unmodified input in the output are **[systematic](https://en.wikipedia.org/wiki/Systematic_code "Systematic code")**, while those that do not are **non-systematic**.
A simplistic example of ECC is to transmit each data bit three times, which is known as a (3,1) [repetition code](https://en.wikipedia.org/wiki/Repetition_code "Repetition code"). Through a noisy channel, a receiver might see eight versions of the output; see the table below.
| Triplet received | Interpreted as |
|---|---|
| 000 | 0 (error-free) |
| 001 | 0 |
| 010 | 0 |
| 100 | 0 |
| 111 | 1 (error-free) |
| 110 | 1 |
| 101 | 1 |
| 011 | 1 |
This allows an error in any one of the three samples to be corrected by "majority vote" or "democratic voting". The correcting ability of this ECC is:
- up to one bit of triplet in error, or
- up to two bits of triplet omitted (cases not shown in table).
Though simple to implement and widely used, this [triple modular redundancy](https://en.wikipedia.org/wiki/Triple_modular_redundancy "Triple modular redundancy") is a relatively inefficient ECC. Better ECC codes typically examine the last several tens or even the last several hundreds of previously received bits to determine how to decode the current small handful of bits (typically in groups of two to eight bits).
### Simplified formalism
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=2 "Edit section: Simplified formalism")\]
Formally, an error-correcting code is given by its ([injective](https://en.wikipedia.org/wiki/Injective_function "Injective function")) encoding function  which assigns to each word  of a finite [alphabet](https://en.wikipedia.org/wiki/Alphabet_\(formal_languages\) "Alphabet (formal languages)")  a unique word  (a concatenation of letters) from the [alphabet](https://en.wikipedia.org/wiki/Alphabet_\(formal_languages\) "Alphabet (formal languages)") .
Most commonly,  is a [homomorphism](https://en.wikipedia.org/wiki/Homomorphism "Homomorphism") in the sense that if  is the concatenation of  and , then we have the following:This implies that it is enough to define  for single-letter words . The [range](https://en.wikipedia.org/wiki/Range_of_a_function "Range of a function")  of the function  is the set of *code-words.* The capabilities of the code to detect and correct errors can then be understood from the *distance*  of the code, which is the minimum [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance "Hamming distance") separating any two distinct code words. A code with distance  can detect errors on  bits as long as , and among those detected errors, the code can correct \-bit errors whenever .
### Averaging noise to reduce errors
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=3 "Edit section: Averaging noise to reduce errors")\]
ECC could be said to work by "averaging noise"; since each data bit affects many transmitted symbols, the corruption of some symbols by noise usually allows the original user data to be extracted from the other, uncorrupted received symbols that also depend on the same user data.
- Because of this "risk-pooling" effect, digital communication systems that use ECC tend to work well above a certain minimum [signal-to-noise ratio](https://en.wikipedia.org/wiki/Signal-to-noise_ratio "Signal-to-noise ratio") and not at all below it.
- This *all-or-nothing tendency* ā the [cliff effect](https://en.wikipedia.org/wiki/Cliff_effect "Cliff effect") ā becomes more pronounced as stronger codes are used that more closely approach the theoretical [Shannon limit](https://en.wikipedia.org/wiki/Shannon_limit "Shannon limit").
- Interleaving ECC coded data can reduce the all or nothing properties of transmitted ECC codes when the channel errors tend to occur in bursts. However, this method has limits; it is best used on narrowband data.
Most telecommunication systems use a fixed [channel code](https://en.wikipedia.org/wiki/Channel_code "Channel code") designed to tolerate the expected worst-case [bit error rate](https://en.wikipedia.org/wiki/Bit_error_rate "Bit error rate"), and then fail to work at all if the bit error rate is ever worse. However, some systems adapt to the given channel error conditions: some instances of [hybrid automatic repeat-request](https://en.wikipedia.org/wiki/Hybrid_automatic_repeat-request "Hybrid automatic repeat-request") use a fixed ECC method as long as the ECC can handle the error rate, then switch to [ARQ](https://en.wikipedia.org/wiki/Automatic_Repeat_Request "Automatic Repeat Request") when the error rate gets too high; [adaptive modulation and coding](https://en.wikipedia.org/wiki/Adaptive_modulation_and_coding "Adaptive modulation and coding") uses a variety of ECC rates, adding more error-correction bits per packet when there are higher error rates in the channel, or taking them out when they are not needed.
[](https://en.wikipedia.org/wiki/File:Block_code_error_correction.png)
A block code (specifically a [Hamming code](https://en.wikipedia.org/wiki/Hamming_code "Hamming code")) where redundant bits are added as a block to the end of the initial message
[](https://en.wikipedia.org/wiki/File:Convolutional_code_error_correction.png)
A continuous [convolutional code](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code") where redundant bits are added continuously into the structure of the code word
The two main categories of ECC codes are [block codes](https://en.wikipedia.org/wiki/Block_code "Block code") and [convolutional codes](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code").
- Block codes work on fixed-size blocks (packets) of bits or symbols of predetermined size. Practical block codes can generally be hard-decoded in [polynomial time](https://en.wikipedia.org/wiki/Polynomial_time "Polynomial time") to their block length.
- Convolutional codes work on bit or symbol streams of arbitrary length. They are most often soft decoded with the [Viterbi algorithm](https://en.wikipedia.org/wiki/Viterbi_algorithm "Viterbi algorithm"), though other algorithms are sometimes used. Viterbi decoding allows asymptotically optimal decoding efficiency with increasing constraint length of the convolutional code, but at the expense of [exponentially](https://en.wikipedia.org/wiki/Exponential_time "Exponential time") increasing complexity. A convolutional code that is terminated is also a 'block code' in that it encodes a block of input data, but the block size of a convolutional code is generally arbitrary, while block codes have a fixed size dictated by their algebraic characteristics. Types of termination for convolutional codes include "tail-biting" and "bit-flushing".
Classical block codes are usually decoded using **hard-decision** algorithms,[\[5\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-5) which means that for every input and output signal a hard decision is made whether it corresponds to a one or a zero bit. In contrast, convolutional codes are typically decoded using **soft-decision** algorithms like the Viterbi, MAP or [BCJR](https://en.wikipedia.org/wiki/BCJR_algorithm "BCJR algorithm") algorithms, which process (discretized) analog signals, and which allow for much higher error-correction performance than hard-decision decoding.
Nearly all classical block codes apply the algebraic properties of [finite fields](https://en.wikipedia.org/wiki/Finite_field "Finite field"). Hence classical block codes are often referred to as algebraic codes.
There are many types of block codes; [ReedāSolomon coding](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction "ReedāSolomon error correction") is noteworthy for its widespread use in [compact discs](https://en.wikipedia.org/wiki/Compact_disc "Compact disc"), [DVDs](https://en.wikipedia.org/wiki/DVD "DVD"), and [hard disk drives](https://en.wikipedia.org/wiki/Hard_disk_drive#Error_rates_and_handling "Hard disk drive"). Other examples of classical block codes include [Golay](https://en.wikipedia.org/wiki/Golay_code_\(disambiguation\) "Golay code (disambiguation)"), [BCH](https://en.wikipedia.org/wiki/BCH_code "BCH code"), [Multidimensional parity](https://en.wikipedia.org/wiki/Multidimensional_parity-check_code "Multidimensional parity-check code"), and [Hamming codes](https://en.wikipedia.org/wiki/Hamming_code "Hamming code").
Hamming ECC is commonly used to correct [ECC memory](https://en.wikipedia.org/wiki/ECC_memory "ECC memory") and early SLC [NAND flash](https://en.wikipedia.org/wiki/NAND_flash "NAND flash") memory errors.[\[6\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-6) This provides single-bit error correction and 2-bit error detection. Hamming codes are only suitable for more reliable [single-level cell](https://en.wikipedia.org/wiki/Single-level_cell "Single-level cell") (SLC) NAND. Denser [multi-level cell](https://en.wikipedia.org/wiki/Multi-level_cell "Multi-level cell") (MLC) NAND may use multi-bit correcting ECC such as [BCH](https://en.wikipedia.org/wiki/BCH_code "BCH code"), ReedāSolomon, or [LDPC](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code").[\[7\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-spansion-7)[\[8\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-8) NOR flash typically does not use any error correction.[\[7\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-spansion-7)
#### Low-density parity-check (LDPC)
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=7 "Edit section: Low-density parity-check (LDPC)")\]
[Low-density parity-check](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code") (LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the [channel capacity](https://en.wikipedia.org/wiki/Shannon_limit "Shannon limit") (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length. Practical implementations rely heavily on decoding the constituent SPC codes in parallel.
LDPC codes were first introduced by [Robert G. Gallager](https://en.wikipedia.org/wiki/Robert_G._Gallager "Robert G. Gallager") in his PhD thesis in 1960, but due to the computational effort in implementing encoder and decoder and the introduction of [ReedāSolomon](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon "ReedāSolomon") codes, they were mostly ignored until the 1990s.
LDPC codes are now used in many recent high-speed communication standards, such as [DVB-S2](https://en.wikipedia.org/wiki/DVB-S2 "DVB-S2") (Digital Video Broadcasting ā Satellite ā Second Generation), [WiMAX](https://en.wikipedia.org/wiki/WiMAX "WiMAX") ([IEEE 802.16e](https://en.wikipedia.org/wiki/IEEE_802.16e "IEEE 802.16e") standard for microwave communications), High-Speed Wireless LAN ([IEEE 802.11n](https://en.wikipedia.org/wiki/IEEE_802.11n "IEEE 802.11n")),[\[9\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-9) [10GBase-T Ethernet](https://en.wikipedia.org/wiki/802.3an#10GBASE-T "802.3an") (802.3an) and [G.hn/G.9960](https://en.wikipedia.org/wiki/G.hn "G.hn") (ITU-T Standard for networking over power lines, phone lines and coaxial cable). Other LDPC codes are standardized for wireless communication standards within [3GPP](https://en.wikipedia.org/wiki/3GPP "3GPP") [MBMS](https://en.wikipedia.org/wiki/MBMS "MBMS") (see [fountain codes](https://en.wikipedia.org/wiki/Fountain_code#Fountain_codes_in_standards "Fountain code")).
[Turbo coding](https://en.wikipedia.org/wiki/Turbo_code "Turbo code") is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the [Shannon limit](https://en.wikipedia.org/wiki/Shannon_limit "Shannon limit"). Predating [LDPC codes](https://en.wikipedia.org/wiki/LDPC_codes "LDPC codes") in terms of practical application, they now provide similar performance.
One of the earliest commercial applications of turbo coding was the [CDMA2000 1x](https://en.wikipedia.org/wiki/CDMA2000 "CDMA2000") (TIA IS-2000) digital cellular technology developed by [Qualcomm](https://en.wikipedia.org/wiki/Qualcomm "Qualcomm") and sold by [Verizon Wireless](https://en.wikipedia.org/wiki/Verizon_Wireless "Verizon Wireless"), [Sprint](https://en.wikipedia.org/wiki/Sprint_Nextel "Sprint Nextel"), and other carriers. It is also used for the evolution of CDMA2000 1x specifically for Internet access, [1xEV-DO](https://en.wikipedia.org/wiki/Evolution-Data_Optimized "Evolution-Data Optimized") (TIA IS-856). Like 1x, EV-DO was developed by [Qualcomm](https://en.wikipedia.org/wiki/Qualcomm "Qualcomm"), and is sold by [Verizon Wireless](https://en.wikipedia.org/wiki/Verizon_Wireless "Verizon Wireless"), [Sprint](https://en.wikipedia.org/wiki/Sprint_Nextel "Sprint Nextel"), and other carriers (Verizon's marketing name for 1xEV-DO is *Broadband Access*, Sprint's consumer and business marketing names for 1xEV-DO are *Power Vision* and *Mobile Broadband*, respectively).
## Describing the performance of an ECC
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=9 "Edit section: Describing the performance of an ECC")\]
In contrast to classical block codes that often specify an error-detecting or error-correcting ability, many modern block codes such as [LDPC codes](https://en.wikipedia.org/wiki/LDPC_codes "LDPC codes") lack such guarantees. Instead, modern codes are evaluated in terms of their bit error rates.
Most forward error correction codes correct only bit-flips, but not bit-insertions or bit-deletions. In this setting, the [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance "Hamming distance") is the appropriate way to measure the [bit error rate](https://en.wikipedia.org/wiki/Bit_error_rate "Bit error rate"). A few forward error correction codes are designed to correct bit-insertions and bit-deletions, such as Marker Codes and Watermark Codes. The [Levenshtein distance](https://en.wikipedia.org/wiki/Levenshtein_distance "Levenshtein distance") is a more appropriate way to measure the bit error rate when using such codes. [\[10\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-10)
### Code-rate and the tradeoff between reliability and data rate
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=10 "Edit section: Code-rate and the tradeoff between reliability and data rate")\]
The fundamental principle of ECC is to add redundant bits in order to help the decoder to find out the true message that was encoded by the transmitter. The code-rate of a given ECC system is defined as the ratio between the number of information bits and the total number of bits (i.e., information plus redundancy bits) in a given communication package. The code-rate is hence a real number. A low code-rate close to zero implies a strong code that uses many redundant bits to achieve a good performance, while a large code-rate close to 1 implies a weak code.
The redundant bits that protect the information have to be transferred using the same communication resources that they are trying to protect. This causes a fundamental tradeoff between reliability and data rate.[\[11\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-11) In one extreme, a strong code (with low code-rate) can induce an important increase in the receiver SNR (signal-to-noise-ratio) decreasing the bit error rate, at the cost of reducing the effective data rate. On the other extreme, not using any ECC (i.e., a code-rate equal to 1) uses the full channel for information transfer purposes, at the cost of leaving the bits without any additional protection.
One interesting question is the following: how efficient in terms of information transfer can an ECC be that has a negligible decoding error rate? This question was answered by Claude Shannon with his second theorem, which says that the channel capacity is the maximum bit rate achievable by any ECC whose error rate tends to zero:[\[12\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-shannon_paper-12) His proof relies on Gaussian random coding, which is not suitable to real-world applications. The upper bound given by Shannon's work inspired a long journey in designing ECCs that can come close to the ultimate performance boundary. Various codes today can attain almost the Shannon limit. However, capacity achieving ECCs are usually extremely complex to implement.
The most popular ECCs have a trade-off between performance and computational complexity. Usually, their parameters give a range of possible code rates, which can be optimized depending on the scenario. Usually, this optimization is done in order to achieve a low decoding error probability while minimizing the impact to the data rate. Another criterion for optimizing the code rate is to balance low error rate and retransmissions number in order to the energy cost of the communication.[\[13\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-13)
## Local decoding and testing of codes
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=11 "Edit section: Local decoding and testing of codes")\]
Sometimes it is only necessary to decode single bits of the message, or to check whether a given signal is a codeword, and do so without looking at the entire signal. This can make sense in a streaming setting, where codewords are too large to be classically decoded fast enough and where only a few bits of the message are of interest for now. Also such codes have become an important tool in [computational complexity theory](https://en.wikipedia.org/wiki/Computational_complexity_theory "Computational complexity theory"), e.g., for the design of [probabilistically checkable proofs](https://en.wikipedia.org/wiki/Probabilistically_checkable_proof "Probabilistically checkable proof").
[Locally decodable codes](https://en.wikipedia.org/wiki/Locally_decodable_code "Locally decodable code") are error-correcting codes for which single bits of the message can be probabilistically recovered by only looking at a small (say constant) number of positions of a codeword, even after the codeword has been corrupted at some constant fraction of positions. [Locally testable codes](https://en.wikipedia.org/wiki/Locally_testable_code "Locally testable code") are error-correcting codes for which it can be checked probabilistically whether a signal is close to a codeword by only looking at a small number of positions of the signal.
Not all locally decodable codes (LDCs) are locally testable codes (LTCs)[\[14\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-decNotTest-14) neither locally correctable codes (LCCs),[\[15\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-15) q-query LCCs are bounded exponentially[\[16\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-16) while LDCs can have [subexponential](https://en.wikipedia.org/wiki/Subexponential_time "Subexponential time") lengths.[\[17\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-17)
## Improving performance
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=12 "Edit section: Improving performance")\]
### Concatenation (combination)
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=13 "Edit section: Concatenation (combination)")\]
Classical (algebraic) block codes and convolutional codes are frequently combined in **concatenated** coding schemes in which a short constraint-length Viterbi-decoded convolutional code does most of the work and a block code (usually ReedāSolomon) with larger symbol size and block length "mops up" any errors made by the convolutional decoder. Single pass decoding with this family of error correction codes can yield very low error rates, but for long range transmission conditions (like deep space) iterative decoding is recommended.
Concatenated codes have been standard practice in satellite and deep space communications since [Voyager 2](https://en.wikipedia.org/wiki/Voyager_program "Voyager program") first used the technique in its 1986 encounter with [Uranus](https://en.wikipedia.org/wiki/Uranus "Uranus"). The [Galileo](https://en.wikipedia.org/wiki/Galileo_\(spacecraft\) "Galileo (spacecraft)") craft used iterative concatenated codes to compensate for the very high error rate conditions caused by having a failed antenna.
"Interleaver" redirects here. For the fiber-optic device, see [optical interleaver](https://en.wikipedia.org/wiki/Optical_interleaver "Optical interleaver").
[](https://en.wikipedia.org/wiki/File:Interleaving1.png)
A short illustration of the interleaving idea
Interleaving is frequently used in digital communication and storage systems to improve the performance of forward error correcting codes. Many [communication channels](https://en.wikipedia.org/wiki/Communication_channel "Communication channel") are not memoryless: errors typically occur in [bursts](https://en.wikipedia.org/wiki/Burst_error "Burst error") rather than independently. If the number of errors within a [code word](https://en.wikipedia.org/wiki/Code_word_\(communication\) "Code word (communication)") exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving alleviates this problem by shuffling source symbols across several code words, thereby creating a more [uniform distribution](https://en.wikipedia.org/wiki/Uniform_distribution_\(continuous\) "Uniform distribution (continuous)") of errors.[\[18\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-turbo-principles-18) Therefore, interleaving is widely used for [burst error-correction](https://en.wikipedia.org/wiki/Burst_error-correcting_code "Burst error-correcting code").
The analysis of modern iterated codes, like [turbo codes](https://en.wikipedia.org/wiki/Turbo_code "Turbo code") and [LDPC codes](https://en.wikipedia.org/wiki/LDPC_code "LDPC code"), typically assumes an independent distribution of errors.[\[19\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-19) Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word.[\[20\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-20)
For turbo codes, an interleaver is an integral component and its proper design is crucial for good performance.[\[18\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-turbo-principles-18)[\[21\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-21) The iterative decoding algorithm works best when there are not short cycles in the [factor graph](https://en.wikipedia.org/wiki/Factor_graph "Factor graph") that represents the decoder; the interleaver is chosen to avoid short cycles.
Interleaver designs include:
- rectangular (or uniform) interleavers (similar to the method using skip factors described above)
- convolutional interleavers
- random interleavers (where the interleaver is a known random permutation)
- S-random interleaver (where the interleaver is a known random permutation with the constraint that no input symbols within distance S appear within a distance of S in the output).[\[22\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-22)
- a contention-free quadratic [permutation polynomial](https://en.wikipedia.org/wiki/Permutation_polynomial "Permutation polynomial") (QPP).[\[23\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-Takeshita1-23) An example of use is in the [3GPP Long Term Evolution](https://en.wikipedia.org/wiki/3GPP_Long_Term_Evolution "3GPP Long Term Evolution") mobile telecommunication standard.[\[24\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-24)
In multi-[carrier](https://en.wikipedia.org/wiki/Carrier_signal "Carrier signal") communication systems, interleaving across carriers may be employed to provide frequency [diversity](https://en.wikipedia.org/wiki/Diversity_scheme "Diversity scheme"), e.g., to mitigate [frequency-selective fading](https://en.wikipedia.org/wiki/Frequency-selective_fading "Frequency-selective fading") or narrowband interference.[\[25\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-25)
#### Interleaving example
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=15 "Edit section: Interleaving example")\]
**Transmission without interleaving**:
```
Error-free message: aaaabbbbccccddddeeeeffffgggg
Transmission with a burst error: aaaabbbbccc____deeeeffffgggg
```
Here, each group of the same letter represents a 4-bit one-bit error-correcting codeword. The codeword cccc is altered in one bit and can be corrected, but the codeword dddd is altered in three bits, so either it cannot be decoded at all or it might be [decoded incorrectly](https://en.wikipedia.org/wiki/Falsing "Falsing").
**With interleaving**:
```
Error-free code words: aaaabbbbccccddddeeeeffffgggg
Interleaved: abcdefgabcdefgabcdefgabcdefg
Transmission with a burst error: abcdefgabcd____bcdefgabcdefg
Received code words after deinterleaving: aa_abbbbccccdddde_eef_ffg_gg
```
In each of the codewords "aaaa", "eeee", "ffff", and "gggg", only one bit is altered, so one-bit error-correcting code will decode everything correctly.
**Transmission without interleaving**:
```
Original transmitted sentence: ThisIsAnExampleOfInterleaving
Received sentence with a burst error: ThisIs______pleOfInterleaving
```
The term "AnExample" ends up mostly unintelligible and difficult to correct.
**With interleaving**:
```
Transmitted sentence: ThisIsAnExampleOfInterleaving...
Error-free transmission: TIEpfeaghsxlIrv.iAaenli.snmOten.
Received sentence with a burst error: TIEpfe______Irv.iAaenli.snmOten.
Received sentence after deinterleaving: T_isI_AnE_amp_eOfInterle_vin_...
```
No word is completely lost and the missing letters can be recovered with minimal guesswork.
#### Disadvantages of interleaving
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=16 "Edit section: Disadvantages of interleaving")\]
Use of interleaving techniques increases total delay. This is because the entire interleaved block must be received before the packets can be decoded.[\[26\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-26) Also interleavers hide the structure of errors; without an interleaver, more advanced decoding algorithms can take advantage of the error structure and achieve more reliable communication than a simpler decoder combined with an interleaver\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]. An example of such an algorithm is based on [neural network](https://en.wikipedia.org/wiki/Neural_network "Neural network")[\[27\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-27) structures.
## Software for error-correcting codes
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=17 "Edit section: Software for error-correcting codes")\]
Simulating the behaviour of error-correcting codes (ECCs) in software is a common practice to design, validate and improve ECCs. The upcoming wireless 5G standard raises a new range of applications for the software ECCs: the [Cloud Radio Access Networks (C-RAN)](https://en.wikipedia.org/wiki/C-RAN "C-RAN") in a [Software-defined radio (SDR)](https://en.wikipedia.org/wiki/Software-defined_radio "Software-defined radio") context. The idea is to directly use software ECCs in the communications. For instance in the 5G, the software ECCs could be located in the cloud and the antennas connected to this computing resources: improving this way the flexibility of the communication network and eventually increasing the energy efficiency of the system.
In this context, there are various available Open-source software listed below (non exhaustive).
- [AFF3CT](https://aff3ct.github.io/)(A Fast Forward Error Correction Toolbox): a full communication chain in C++ (many supported codes like Turbo, LDPC, Polar codes, etc.), very fast and specialized on channel coding (can be used as a program for simulations or as a library for the SDR).
- [IT++](https://en.wikipedia.org/wiki/IT%2B%2B "IT++"): a C++ library of classes and functions for linear algebra, numerical optimization, signal processing, communications, and statistics.
- [OpenAir](https://gitlab.eurecom.fr/oai/openair-cn): implementation (in C) of the 3GPP specifications concerning the Evolved Packet Core Networks.
## List of error-correcting codes
\[[edit](https://en.wikipedia.org/w/index.php?title=Error_correction_code&action=edit§ion=18 "Edit section: List of error-correcting codes")\]
| Code | Distance | Detectable errors (bits) | Correctable errors (bits) |
|---|---|---|---|
| Parity (guess needed on error) | 2 | 1 | 0 |
| [Triple modular redundancy](https://en.wikipedia.org/wiki/Triple_modular_redundancy "Triple modular redundancy") | 3 | 2 | 1 |
| Perfect Hamming such as [Hamming(7,4)](https://en.wikipedia.org/wiki/Hamming\(7,4\) "Hamming(7,4)") | 3 | 2 | 1 |
| [SECDED](https://en.wikipedia.org/wiki/SECDED "SECDED"): extended Hamming such as (39,32), (72,64) | 4 | 3 | 1 |
| DECTED: [Nordstrom-Robinson code](https://en.wikipedia.org/wiki/Nordstrom-Robinson_code "Nordstrom-Robinson code") | 6 | 5 | 2 |
| Perfect [binary Golay code](https://en.wikipedia.org/wiki/Binary_Golay_code "Binary Golay code") | 7 | 6 | 3 |
| TECFED: Extended [binary Golay code](https://en.wikipedia.org/wiki/Binary_Golay_code "Binary Golay code") | 8 | 7 | 3 |
In the table above, for an error-correcting code of minimal Hamming distance , the maximum number of errors that the code can detect is given by  while the maximal number of errors that it can correct is given by .
- [AN codes](https://en.wikipedia.org/wiki/AN_codes "AN codes")
- [Algebraic geometry code](https://en.wikipedia.org/wiki/Algebraic_geometry_code "Algebraic geometry code")
- [BCH code](https://en.wikipedia.org/wiki/BCH_code "BCH code"), which can be designed to correct any arbitrary number of errors per code block.
- [Barker code](https://en.wikipedia.org/wiki/Barker_code "Barker code") used for radar, telemetry, ultra sound, Wifi, DSSS mobile phone networks, GPS etc.
- [Berger code](https://en.wikipedia.org/wiki/Berger_code "Berger code")
- [Constant-weight code](https://en.wikipedia.org/wiki/Constant-weight_code "Constant-weight code")
- [Convolutional code](https://en.wikipedia.org/wiki/Convolutional_code "Convolutional code")
- [Expander codes](https://en.wikipedia.org/wiki/Expander_code "Expander code")
- [Group codes](https://en.wikipedia.org/wiki/Group_code "Group code")
- [Golay codes](https://en.wikipedia.org/wiki/Golay_code_\(disambiguation\) "Golay code (disambiguation)"), of which the [Binary Golay code](https://en.wikipedia.org/wiki/Binary_Golay_code "Binary Golay code") is of practical interest
- [Goppa code](https://en.wikipedia.org/wiki/Binary_Goppa_code "Binary Goppa code"), used in the [McEliece cryptosystem](https://en.wikipedia.org/wiki/McEliece_cryptosystem "McEliece cryptosystem")
- [Hadamard code](https://en.wikipedia.org/wiki/Hadamard_code "Hadamard code")
- [Hagelbarger code](https://en.wikipedia.org/wiki/Hagelbarger_code "Hagelbarger code")
- [Hamming code](https://en.wikipedia.org/wiki/Hamming_code "Hamming code")
- [Latin square based code](https://en.wikipedia.org/wiki/Latin_square#Error_correcting_codes "Latin square") for non-white noise (prevalent for example in broadband over powerlines)
- [Lexicographic code](https://en.wikipedia.org/wiki/Lexicographic_code "Lexicographic code")
- [Linear Network Coding](https://en.wikipedia.org/wiki/Linear_network_coding "Linear network coding"), a type of erasure correcting code across networks instead of point-to-point links
- [Long code](https://en.wikipedia.org/wiki/Long_code_\(mathematics\) "Long code (mathematics)")
- [Low-density parity-check code](https://en.wikipedia.org/wiki/Low-density_parity-check_code "Low-density parity-check code"), also known as [Gallager code](https://en.wikipedia.org/wiki/Gallager_code "Gallager code"), as the archetype for [sparse graph codes](https://en.wikipedia.org/wiki/Sparse_graph_code "Sparse graph code")
- [LT code](https://en.wikipedia.org/wiki/LT_code "LT code"), which is a near-optimal [rateless erasure correcting code (Fountain code)](https://en.wikipedia.org/wiki/Fountain_code "Fountain code")
- [m of n codes](https://en.wikipedia.org/wiki/M_of_n_codes "M of n codes")
- [Nordstrom-Robinson code](https://en.wikipedia.org/wiki/Nordstrom-Robinson_code "Nordstrom-Robinson code"), used in Geometry and Group Theory[\[28\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-28)
- [Online code](https://en.wikipedia.org/wiki/Online_code "Online code"), a near-optimal [rateless erasure correcting code](https://en.wikipedia.org/wiki/Fountain_code "Fountain code")
- [Polar code (coding theory)](https://en.wikipedia.org/wiki/Polar_code_\(coding_theory\) "Polar code (coding theory)")
- [Raptor code](https://en.wikipedia.org/wiki/Raptor_code "Raptor code"), a near-optimal [rateless erasure correcting code](https://en.wikipedia.org/wiki/Fountain_code "Fountain code")
- [ReedāSolomon error correction](https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction "ReedāSolomon error correction")
- [ReedāMuller code](https://en.wikipedia.org/wiki/Reed%E2%80%93Muller_code "ReedāMuller code")
- [Repeat-accumulate code](https://en.wikipedia.org/wiki/Repeat-accumulate_code "Repeat-accumulate code")
- [Repetition codes](https://en.wikipedia.org/wiki/Repetition_code "Repetition code"), such as [Triple modular redundancy](https://en.wikipedia.org/wiki/Triple_modular_redundancy "Triple modular redundancy")
- Spinal code, a rateless, nonlinear code based on pseudo-random hash functions[\[29\]](https://en.wikipedia.org/wiki/Error_correction_code#cite_note-29)
- [Tornado code](https://en.wikipedia.org/wiki/Tornado_code "Tornado code"), a near-optimal [erasure correcting code](https://en.wikipedia.org/wiki/Erasure_code "Erasure code"), and the precursor to [Fountain codes](https://en.wikipedia.org/wiki/Fountain_code "Fountain code")
- [Turbo code](https://en.wikipedia.org/wiki/Turbo_code "Turbo code")
- [WalshāHadamard code](https://en.wikipedia.org/wiki/Walsh%E2%80%93Hadamard_code "WalshāHadamard code")
- [Cyclic redundancy checks](https://en.wikipedia.org/wiki/Cyclic_redundancy_check "Cyclic redundancy check") (CRCs) can correct 1-bit errors for messages at most  bits long for optimal generator polynomials of degree , see [Mathematics of cyclic redundancy checks Ā§ Bitfilters](https://en.wikipedia.org/wiki/Mathematics_of_cyclic_redundancy_checks#Bitfilters "Mathematics of cyclic redundancy checks")
- [Locally Recoverable Codes](https://en.wikipedia.org/wiki/Locally_Recoverable_Codes "Locally Recoverable Codes")
- [Message authentication code](https://en.wikipedia.org/wiki/Message_authentication_code "Message authentication code")
- [Burst error-correcting code](https://en.wikipedia.org/wiki/Burst_error-correcting_code "Burst error-correcting code")
- [Code rate](https://en.wikipedia.org/wiki/Code_rate "Code rate")
- [Erasure codes](https://en.wikipedia.org/wiki/Erasure_code "Erasure code")
- [Error detection and correction](https://en.wikipedia.org/wiki/Error_detection_and_correction "Error detection and correction")
- [Error-correcting codes with feedback](https://en.wikipedia.org/wiki/Error-correcting_codes_with_feedback "Error-correcting codes with feedback")
- [Linear code](https://en.wikipedia.org/wiki/Linear_code "Linear code")
- [Quantum error correction](https://en.wikipedia.org/wiki/Quantum_error_correction "Quantum error correction")
- [Soft-decision decoder](https://en.wikipedia.org/wiki/Soft-decision_decoder "Soft-decision decoder")
1. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-1)**
Charles Wang; Dean Sklar; Diana Johnson (Winter 2001ā2002). ["Forward Error-Correction Coding"](https://web.archive.org/web/20120314085127/http://www.aero.org/publications/crosslink/winter2002/04.html). *Crosslink*. **3** (1). The Aerospace Corporation. Archived from [the original](http://www.aero.org/publications/crosslink/winter2002/04.html) on 14 March 2012. Retrieved 5 March 2006.
2. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-2)**
Glover, Neal; Dudley, Trent (1990). *Practical Error Correction Design For Engineers* (Revision 1.1, 2nd ed.). CO, USA: [Cirrus Logic](https://en.wikipedia.org/wiki/Cirrus_Logic "Cirrus Logic"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-927239-00-0](https://en.wikipedia.org/wiki/Special:BookSources/0-927239-00-0 "Special:BookSources/0-927239-00-0")
.
3. ^ [***a***](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-Hamming_3-0) [***b***](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-Hamming_3-1)
[Hamming, Richard Wesley](https://en.wikipedia.org/wiki/Richard_Wesley_Hamming "Richard Wesley Hamming") (April 1950). "Error Detecting and Error Correcting Codes". *[Bell System Technical Journal](https://en.wikipedia.org/wiki/Bell_System_Technical_Journal "Bell System Technical Journal")*. **29** (2). USA: [AT\&T](https://en.wikipedia.org/wiki/AT%26T "AT&T"): 147ā160\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1950BSTJ...29..147H](https://ui.adsabs.harvard.edu/abs/1950BSTJ...29..147H). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/j.1538-7305.1950.tb00463.x](https://doi.org/10.1002%2Fj.1538-7305.1950.tb00463.x). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[10945/46756](https://hdl.handle.net/10945%2F46756). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [61141773](https://api.semanticscholar.org/CorpusID:61141773).
4. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-:0_4-0)**
Maunder, Robert (2016). ["Overview of Channel Coding"](https://www.accelercomm.com/overview-channel-coding).
5. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-5)**
Baldi, M.; Chiaraluce, F. (2008). ["A Simple Scheme for Belief Propagation Decoding of BCH and RS Codes in Multimedia Transmissions"](https://doi.org/10.1155%2F2008%2F957846). *[International Journal of Digital Multimedia Broadcasting](https://en.wikipedia.org/wiki/International_Journal_of_Digital_Multimedia_Broadcasting "International Journal of Digital Multimedia Broadcasting")*. **2008**: 1ā12\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1155/2008/957846](https://doi.org/10.1155%2F2008%2F957846).
6. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-6)** ["Hamming codes for NAND flash memory devices"](http://www.eetasia.com/ART_8800575062_499486_AN_7549c493.HTM) [Archived](https://web.archive.org/web/20160821122453/http://www.eetasia.com/ART_8800575062_499486_AN_7549c493.HTM) 21 August 2016 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine"). EE Times-Asia. Apparently based on ["Micron Technical Note TN-29-08: Hamming Codes for NAND Flash Memory Devices"](http://www.micron.com/~/media/Documents/Products/Technical%20Note/NAND%20Flash/tn2908_NAND_hamming_ECC_code.pdf) [Archived](https://web.archive.org/web/20170829073235/http://www.micron.com/~/media/Documents/Products/Technical%20Note/NAND%20Flash/tn2908_NAND_hamming_ECC_code.pdf) 29 August 2017 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine"). 2005. Both say: "The Hamming algorithm is an industry-accepted method for error detection and correction in many SLC NAND flash-based applications."
7. ^ [***a***](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-spansion_7-0) [***b***](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-spansion_7-1)
["What Types of ECC Should Be Used on Flash Memory?"](https://web.archive.org/web/20160614220343/http://www.spansion.com/Support/Application%20Notes/Types_of_ECC_Used_on_Flash_AN.pdf) (PDF). Spansion. 2011. Archived from [the original](http://www.spansion.com/Support/Application%20Notes/Types_of_ECC_Used_on_Flash_AN.pdf) (Application note) on 14 June 2016. "Both ReedāSolomon algorithm and BCH algorithm are common ECC choices for MLC NAND flash. ... Hamming based block codes are the most commonly used ECC for SLC.... both ReedāSolomon and BCH are able to handle multiple errors and are widely used on MLC flash."
8. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-8)**
Jim Cooke (August 2007). ["The Inconvenient Truths of NAND Flash Memory"](https://cushychicken.github.io/assets/cooke_inconvenient_truths.pdf) (PDF). p. 28. "For SLC, a code with a correction threshold of 1 is sufficient. t=4 required ... for MLC."
9. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-9)** IEEE Standard, section 20.3.11.6 ["802.11n-2009"](http://standards.ieee.org/getieee802/download/802.11n-2009.pdf) [Archived](https://web.archive.org/web/20130203104520/http://standards.ieee.org/getieee802/download/802.11n-2009.pdf) 3 February 2013 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine"), IEEE, 29 October 2009, accessed 21 March 2011.
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Shah, Gaurav; Molina, Andres; Blaze, Matt (2006). ["Keyboards and covert channels"](https://www.usenix.org/legacy/event/sec06/tech/full_papers/shah/shah_html/jbug-Usenix06.html). *USENIX*. Retrieved 20 December 2018.
11. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-11)**
Tse, David; Viswanath, Pramod (2005), *Fundamentals of Wireless Communication*, [Cambridge University Press](https://en.wikipedia.org/wiki/Cambridge_University_Press "Cambridge University Press"), UK
12. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-shannon_paper_12-0)**
Shannon, C. E. (1948). ["A mathematical theory of communication"](http://www.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf) (PDF). *[Bell System Technical Journal](https://en.wikipedia.org/wiki/Bell_System_Technical_Journal "Bell System Technical Journal")*. **27** (3ā4\): 379ā423 & 623ā656. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1948BSTJ...27..379S](https://ui.adsabs.harvard.edu/abs/1948BSTJ...27..379S). [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)"):[11858/00-001M-0000-002C-4314-2](https://hdl.handle.net/11858%2F00-001M-0000-002C-4314-2).
13. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-13)**
Rosas, F.; Brante, G.; Souza, R. D.; Oberli, C. (2014). "Optimizing the code rate for achieving energy-efficient wireless communications". *Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC)*. pp. 775ā780\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/WCNC.2014.6952166](https://doi.org/10.1109%2FWCNC.2014.6952166). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-4799-3083-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4799-3083-8 "Special:BookSources/978-1-4799-3083-8")
.
14. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-decNotTest_14-0)**
[Kaufman, Tali](https://en.wikipedia.org/wiki/Tali_Kaufman "Tali Kaufman"); Viderman, Michael. ["Locally Testable vs. Locally Decodable Codes"](http://eccc.hpi-web.de/report/2010/130/revision/1/download/).
15. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-15)**
Brubaker, Ben (9 January 2024). ["'Magical' Error Correction Scheme Proved Inherently Inefficient"](https://www.quantamagazine.org/magical-error-correction-scheme-proved-inherently-inefficient-20240109/). *Quanta Magazine*. Retrieved 9 January 2024.
16. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-16)**
Kothari, Pravesh; Manohar, Peter (11 June 2024). ["An Exponential Lower Bound for Linear 3-Query Locally Correctable Codes"](https://dl.acm.org/doi/epdf/10.1145/3618260.3649640). *Symposium on Theory of Computing*. **56**: 776ā787\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1145/3618260.3649640](https://doi.org/10.1145%2F3618260.3649640). Retrieved 24 March 2026.
17. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-17)**
Efremenko, Klim (31 May 2009). ["3-query locally decodable codes of subexponential length"](https://epubs.siam.org/doi/10.1137/090772721). *SIAM Journal on Computing*. **41** (6): 39ā44\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1145/1536414.1536422](https://doi.org/10.1145%2F1536414.1536422). Retrieved 24 March 2026.
18. ^ [***a***](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-turbo-principles_18-0) [***b***](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-turbo-principles_18-1)
Vucetic, B.; Yuan, J. (2000). *Turbo codes: principles and applications*. [Springer Verlag](https://en.wikipedia.org/wiki/Springer_Verlag "Springer Verlag"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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.
19. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-19)**
[Luby, Michael](https://en.wikipedia.org/wiki/Michael_Luby "Michael Luby"); Mitzenmacher, M.; Shokrollahi, A.; Spielman, D.; Stemann, V. (1997). "Practical Loss-Resilient Codes". *Proc. 29th Annual Association for Computing Machinery (ACM) Symposium on Theory of Computation*.
20. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-20)**
"Digital Video Broadcast (DVB); Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other satellite broadband applications (DVB-S2)". *En 302 307* (V1.2.1). [ETSI](https://en.wikipedia.org/wiki/ETSI "ETSI"). April 2009.
21. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-21)**
Andrews, K. S.; Divsalar, D.; Dolinar, S.; Hamkins, J.; Jones, C. R.; Pollara, F. (November 2007). "The Development of Turbo and LDPC Codes for Deep-Space Applications". *[Proceedings of the IEEE](https://en.wikipedia.org/wiki/Proceedings_of_the_IEEE "Proceedings of the IEEE")*. **95** (11): 2142ā2156\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/JPROC.2007.905132](https://doi.org/10.1109%2FJPROC.2007.905132). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [9289140](https://api.semanticscholar.org/CorpusID:9289140).
22. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-22)**
Dolinar, S.; Divsalar, D. (15 August 1995). "Weight Distributions for Turbo Codes Using Random and Nonrandom Permutations". *TDA Progress Report*. **122**: 42ā122\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1995TDAPR.122...56D](https://ui.adsabs.harvard.edu/abs/1995TDAPR.122...56D). [CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_\(identifier\) "CiteSeerX (identifier)") [10\.1.1.105.6640](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.105.6640).
23. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-Takeshita1_23-0)**
Takeshita, Oscar (2006). "Permutation Polynomial Interleavers: An Algebraic-Geometric Perspective". *[IEEE Transactions on Information Theory](https://en.wikipedia.org/wiki/IEEE_Transactions_on_Information_Theory "IEEE Transactions on Information Theory")*. **53** (6): 2116ā2132\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[cs/0601048](https://arxiv.org/abs/cs/0601048). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2006cs........1048T](https://ui.adsabs.harvard.edu/abs/2006cs........1048T). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1109/TIT.2007.896870](https://doi.org/10.1109%2FTIT.2007.896870). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [660](https://api.semanticscholar.org/CorpusID:660).
24. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-24)** [3GPP TS 36.212](http://www.3gpp.org/ftp/Specs/html-info/36212.htm), version 8.8.0, page 14
25. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-25)**
"Digital Video Broadcast (DVB); Frame structure, channel coding and modulation for a second generation digital terrestrial television broadcasting system (DVB-T2)". *En 302 755* (V1.1.1). [ETSI](https://en.wikipedia.org/wiki/ETSI "ETSI"). September 2009.
26. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-26)**
Techie (3 June 2010). ["Explaining Interleaving"](http://w3techie.com/2010/explaining-interleaving/). *W3 Techie Blog*. Retrieved 3 June 2010.
27. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-27)**
Krastanov, Stefan; Jiang, Liang (8 September 2017). ["Deep Neural Network Probabilistic Decoder for Stabilizer Codes"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5591216). *Scientific Reports*. **7** (1): 11003. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1705\.09334](https://arxiv.org/abs/1705.09334). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2017NatSR...711003K](https://ui.adsabs.harvard.edu/abs/2017NatSR...711003K). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/s41598-017-11266-1](https://doi.org/10.1038%2Fs41598-017-11266-1). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [5591216](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5591216). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [28887480](https://pubmed.ncbi.nlm.nih.gov/28887480).
28. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-28)**
Nordstrom, A.W.; Robinson, J.P. (1967), "An optimum nonlinear code", *Information and Control*, **11** (5ā6\): 613ā616, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/S0019-9958(67)90835-2](https://doi.org/10.1016%2FS0019-9958%2867%2990835-2)
29. **[^](https://en.wikipedia.org/wiki/Error_correction_code#cite_ref-29)**
Perry, Jonathan; Balakrishnan, Hari; Shah, Devavrat (2011). ["Rateless Spinal Codes"](http://doi.acm.org/10.1145/2070562.2070568). *Proceedings of the 10th ACM Workshop on Hot Topics in Networks*. pp. 1ā6\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1145/2070562.2070568](https://doi.org/10.1145%2F2070562.2070568). [hdl](https://en.wikipedia.org/wiki/Hdl_\(identifier\) "Hdl (identifier)"):[1721\.1/79676](https://hdl.handle.net/1721.1%2F79676). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781450310598](https://en.wikipedia.org/wiki/Special:BookSources/9781450310598 "Special:BookSources/9781450310598")
.
- [MacWilliams, Florence Jessiem](https://en.wikipedia.org/wiki/Florence_Jessiem_MacWilliams "Florence Jessiem MacWilliams"); [Sloane, Neil James Alexander](https://en.wikipedia.org/wiki/Neil_James_Alexander_Sloane "Neil James Alexander Sloane") (2007) \[1977\]. Written at AT\&T Shannon Labs, Florham Park, New Jersey, USA. *The Theory of Error-Correcting Codes*. North-Holland Mathematical Library. Vol. 16 (digital print of 12th impression, 1st ed.). Amsterdam / London / New York / Tokyo: [North-Holland](https://en.wikipedia.org/wiki/North-Holland_Publishing_Company "North-Holland Publishing Company") / [Elsevier BV](https://en.wikipedia.org/wiki/Elsevier_BV "Elsevier BV"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-444-85193-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-444-85193-2 "Special:BookSources/978-0-444-85193-2")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [76-41296](https://lccn.loc.gov/76-41296).
(xxii+762+6 pages)
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[0-262-01098-4](https://en.wikipedia.org/wiki/Special:BookSources/0-262-01098-4 "Special:BookSources/0-262-01098-4")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [87-21889](https://lccn.loc.gov/87-21889).
(x+2+208+4 pages)
- Wicker, Stephen B. (1995). *Error Control Systems for Digital Communication and Storage*. Englewood Cliffs, New Jersey, USA: [Prentice-Hall](https://en.wikipedia.org/wiki/Prentice-Hall "Prentice-Hall"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-13-200809-2](https://en.wikipedia.org/wiki/Special:BookSources/0-13-200809-2 "Special:BookSources/0-13-200809-2")
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- Wilson, Stephen G. (1996). *Digital Modulation and Coding*. Englewood Cliffs, New Jersey, USA: [Prentice-Hall](https://en.wikipedia.org/wiki/Prentice-Hall "Prentice-Hall"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-13-210071-1](https://en.wikipedia.org/wiki/Special:BookSources/0-13-210071-1 "Special:BookSources/0-13-210071-1")
.
- ["Error Correction Code in Single Level Cell NAND Flash memories"](https://web.archive.org/web/20070519161253/http://www.st.com/stonline/products/literature/an/10123.htm) 2007-02-16
- ["Error Correction Code in NAND Flash memories"](https://web.archive.org/web/20160304050018/http://www.eetasia.com/ARTICLES/2004NOV/A/2004NOV29_MEM_AN10.PDF?SOURCES=DOWNLOAD) 2004-11-29
- [Observations on Errors, Corrections, & Trust of Dependent Systems](http://perspectives.mvdirona.com/2012/02/observations-on-errors-corrections-trust-of-dependent-systems/), by James Hamilton, 2012-02-26
- *[Sphere Packings, Lattices and Groups](https://en.wikipedia.org/wiki/Sphere_Packings,_Lattices_and_Groups "Sphere Packings, Lattices and Groups"),* by J. H. Conway, Neil James Alexander Sloane, [Springer Science & Business Media](https://en.wikipedia.org/wiki/Springer_Science_%26_Business_Media "Springer Science & Business Media"), 2013-03-09 ā Mathematics ā 682 pages.
- Morelos-Zaragoza, Robert (2004). ["The Correcting Codes (ECC) Page"](http://www.eccpage.com/). Retrieved 5 March 2006.
- [error correction zoo](https://errorcorrectionzoo.org/). Database of error correcting codes.
- [lpdec: library for LP decoding and related things (Python)](https://github.com/supermihi/lpdec) |
| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Error_correction_code s443 |