Error correction code: Difference between revisions

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{{short description|scheme for controlling errors in data over noisy communication channels}}

{{redirect|Interleaver|the fiber-optic device|optical interleaver}}

{{Use dmy dates|date=July 2013}}

In [[computing]], [[telecommunication]], [[information theory]], and [[coding theory]], an '''error correction code,''' sometimes '''error correcting code,''' ('''ECC''') is used for [[error control|controlling errors]] in data over unreliable or noisy [[communication channel]]s.<ref> {{cite book |author-last1=Glover |author-first1=Neal |author-last2=Dudley |author-first2=Trent |title=Practical Error Correction Design For Engineers |edition=Revision 1.1, 2nd |publisher=[[Cirrus Logic]] |___location=CO, USA |date=1990 |isbn=0-927239-00-0 |id={{ISBN|978-0-927239-00-4}}}}</ref><ref name="Hamming">{{cite journal |author-last=Hamming |author-first=R. W. |title=Error Detecting and Error Correcting Codes |journal=[[Bell System Technical Journal]] |volume=29 |issue=2 |pages=147–160 |publisher=[[AT&T]] |___location=USA |date=April 1950 |url=http://www3.alcatel-lucent.com/bstj/vol29-1950/articles/bstj29-2-147.pdf |doi=10.1002/j.1538-7305.1950.tb00463.x |access-date=4 December 2012}}</ref> The central idea is the sender encodes the message with a [[Redundancy (information theory)|redundant]] in the form of an ECC. 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]].<ref name="Hamming"/>

The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. ECC gives the receiver the ability to correct errors without needing a [[reverse channel]] to request retransmission of data, but at the cost of a fixed, higher forward channel bandwidth. ECC is therefore applied in situations where retransmissions are costly or impossible, such as one-way communication links and when transmitting to multiple receivers in [[multicast]]. For example, in the case of a satellite orbiting around [[Uranus]], a retransmission because of decoding errors can create a delay of 5 hours. ECC information is usually added to [[mass storage]] devices to enable recovery of corrupted data, is widely used in [[modem]]s, and is used on systems where the primary memory is [[ECC memory]].

| last1 = Glover | first1 = Neal

| last2 = Dudley | first2 = Trent

| title = Practical Error Correction Design For Engineers

| edition= Revision 1.1, 2^nd

| publisher = Cirrus Logic

| ___location = CO, USA

| year = 1990

| isbn= 978-0-927239-00-4

}}</ref><ref name="Hamming">{{cite journal

| last = Hamming

| first = R. W.

| title = Error Detecting and Error Correcting Codes

| journal = Bell System Tech. J.

| volume = 29

| issue = 2

| pages = 147–160

| publisher = AT&T

| ___location = USA

| date = April 1950

| url = http://www3.alcatel-lucent.com/bstj/vol29-1950/articles/bstj29-2-147.pdf

| issn =

| doi = 10.1002/j.1538-7305.1950.tb00463.x

| id =

| accessdate = 4 December 2012}}</ref> The central idea is the sender encodes the message with a [[Redundancy (information theory)|redundant]] in the form of an ECC. 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]].<ref name="Hamming" />

The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. ECC gives the receiver the ability to correct errors without needing a [[reverse channel]] to request retransmission of data, but at the cost of a fixed, higher forward channel bandwidth. ECC is therefore applied in situations where retransmissions are costly or impossible, such as one-way communication links and when transmitting to multiple receivers in [[multicast]]. For example, in the case of a satellite orbiting around [[Uranus]], a retransmission because of decoding errors can create a delay of 5 hours. ECC information is usually added to [[mass storage]] devices to enable recovery of corrupted data, is widely used in [[modem]]s, and is used on systems where the primary memory is [[ECC memory]].

ECC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, ECC is an integral part of the initial [[Analog-to-digital converter|analog-to-digital conversion]] in the receiver. The [[Viterbi decoder]] implements a [[Error correction code#Types of ECC|soft-decision algorithm]] to demodulate digital data from an analog signal corrupted by noise. Many ECC encoders/decoders can also generate a [[bit-error rate]] (BER) signal which can be used as feedback to fine-tune the analog receiving electronics.

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There are many types of block codes, but among the classical ones the most notable is [[Reed-Solomon error correction|Reed-Solomon coding]] because of its widespread use in [[compact disc]]s, [[DVD]]s, and [[hard disk drive#Error handling|hard disk drives]]. Other examples of classical block codes include [[Golay code (disambiguation)|Golay]], [[BCH code|BCH]], [[Multidimensional parity-check code|Multidimensional parity]], and [[Hamming code]]s.

Hamming ECC is commonly used to correct [[NAND flash]] memory errors.<ref>[http://www.eetasia.com/ART_8800575062_499486_AN_7549c493.HTM "Hamming codes for NAND flash memory devices"]. EE Times-Asia. Apparently based on [http://www.micron.com/~/media/Documents/Products/Technical%20Note/NAND%20Flash/tn2908_NAND_hamming_ECC_code.pdf "Micron Technical Note TN-29-08: Hamming Codes for NAND Flash Memory Devices"]. 2005. Both say: "The Hamming algorithm is an industry-accepted method for error detection and correction in many SLC NAND flash-based applications."</ref>

Hamming ECC is commonly used to correct [[NAND flash]] memory errors.<ref>

[http://www.eetasia.com/ART_8800575062_499486_AN_7549c493.HTM "Hamming codes for NAND flash memory devices"].

EE Times-Asia.

Apparently based on

[http://www.micron.com/~/media/Documents/Products/Technical%20Note/NAND%20Flash/tn2908_NAND_hamming_ECC_code.pdf "Micron Technical Note TN-29-08: Hamming Codes for NAND Flash Memory Devices"].

2005.

Both say:

"The Hamming algorithm is an industry-accepted method for error detection and correction in many SLC NAND flash-based applications."

</ref>

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 requires stronger multi-bit correcting ECC such as BCH or Reed–Solomon.<ref name="spansion">[http://www.spansion.com/Support/Application%20Notes/Types_of_ECC_Used_on_Flash_AN.pdf "What Types of ECC Should Be Used on Flash Memory?"]. (Spansion application note). 2011. says: "Both Reed-Solomon 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 Reed-Solomon and BCH are able to handle multiple errors and are widely used on MLC flash."</ref><ref>Jim Cooke. {https://cushychicken.github.io/assets/cooke_inconvenient_truths.pdf "The Inconvenient Truths of NAND Flash Memory"]. 2007. p. 28. says "For SLC, a code with a correction threshold of 1 is sufficient. t=4 required ... for MLC."</ref>{{Dubious|date=March 2008}} NOR Flash typically does not use any error correction.<ref name="spansion"/>

Denser [[multi-level cell]] (MLC) NAND requires stronger multi-bit correcting ECC such as BCH or Reed–Solomon.<ref name="spansion">

[http://www.spansion.com/Support/Application%20Notes/Types_of_ECC_Used_on_Flash_AN.pdf "What Types of ECC Should Be Used on Flash Memory?"].

(Spansion application note).

2011.

says: "Both Reed-Solomon 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 Reed-Solomon and BCH are able to handle multiple errors and are widely used on MLC flash."

</ref><ref>

Jim Cooke.

[https://cushychicken.github.io/assets/cooke_inconvenient_truths.pdf "The Inconvenient Truths of NAND Flash Memory"].

2007.

p. 28.

says

"For SLC, a code with a correction threshold of 1 is sufficient. t=4 required ... for MLC."

</ref>

{{Dubious|date=March 2008}}

NOR Flash typically does not use any error correction.<ref name="spansion" />

Classical block codes are usually decoded using '''hard-decision''' algorithms,<ref>{{cite journal |author1author-last1=Baldi |author-first1=M. |author2author-last2=Chiaraluce |author-first2=F. |title=A Simple Scheme for Belief Propagation Decoding of BCH and RS Codes in Multimedia Transmissions |journal=[[International Journal of Digital Multimedia Broadcasting]] |volume=2008 |pages=1–12 |yeardate=2008 |doi=10.1155/2008/957846 |url=http://www.hindawi.com/journals/ijdmb/2008/957846.html}}</ref> 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 algorithm|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 field]]s. Hence classical block codes are often referred to as algebraic codes.

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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.

<ref>{{cite web |author-last1=Shah |author-first1=Gaurav |author-last2=Molina |author-first2=Andres |author-last3=Blaze |author-first3=Matt |title=Keyboards and covert channels |url=https://www.usenix.org/legacy/event/sec06/tech/full_papers/shah/shah_html/jbug-Usenix06.html |website=Usenix.org |access-date=20 December 2018 |date=2006}}</ref>

<ref>{{cite web

| last1=Shah | first1=Gaurav

| last2=Molina | first2=Andres

| last3=Blaze | first3=Matt

| title=Keyboards and covert channels

| url=https://www.usenix.org/legacy/event/sec06/tech/full_papers/shah/shah_html/jbug-Usenix06.html

| website=Usenix.org

| accessdate=20 December 2018

| year=2006

}}

</ref>

==Code-rate and the tradeoff between reliability and data rate==

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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 rate 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.<ref>{{citation | author -first1= David |author-last1=Tse, |author-first2=Pramod |author-last2=Viswanath | title = Fundamentals of Wireless Communication | publisher = [[Cambridge University Press]], UK | yeardate=2005}}</ref>. In one extreme, a strong code (with low code-rate) can induce an important increase in the receiver SNR 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 be an ECC 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:<ref name="shannon paper">C. E. Shannon: ''A mathematical theory of communication.'' [[Bell System Technical Journal]], vol. 27, pp. 379–423 and 623–656, July and October 1948</ref>. His proof relies on Gaussian random coding, which is not suitable of real-world applications. This upper bound given by Shannon's work set up a long journey in designing ECCs that can go 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.<ref>{{Cite conference |title=Optimizing the code rate for achieving energy-efficient wireless communications |first1=Fernando |last1=Rosas |first2=Glauber |last2=Brante |first3=Richard Demo |last3=Souza |first4=Christian |last4=Oberli |url=http://ieeexplore.ieee.org/abstract/document/6952166/ |date=2014 |booktitle=Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC)}}</ref>

{{Cite conference | title= Optimizing the code rate for achieving energy-efficient wireless communications| first1 = Fernando | last1 = Rosas| first2 = Glauber | last2 = Brante| first3 = Richard Demo | last3 = Souza| first4 = Christian | last4 = Oberli| url = http://ieeexplore.ieee.org/abstract/document/6952166/| year= 2014| booktitle=Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC)}}</ref>.

==Concatenated ECC codes for improved performance==

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[[File:Interleaving1.png|right|400px|thumb|A short illustration of interleaving idea.]]

Interleaving is frequently used in digital communication and storage systems to improve the performance of forward error correcting codes. Many [[communication channel]]s are not memoryless: errors typically occur in [[burst error|burst]]s 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 ameliorates this problem by shuffling source symbols across several code words, thereby creating a more [[Uniform distribution (continuous)|uniform distribution]] of errors.<ref name="turbo-principles">{{cite book |author1author-first1=B. |author-last1=Vucetic |author2author-first2=J. |author-last2=Yuan |title=Turbo codes: principles and applications |publisher=[[Springer Verlag]] |isbn=978-0-7923-7868-6 |yeardate=2000}}</ref> Therefore, interleaving is widely used for [[burst error-correcting code|burst error-correction]].

The analysis of modern iterated codes, like [[turbo code]]s and [[LDPC code]]s, typically assumes an independent distribution of errors.<ref>{{cite journal |author-first1=[[Michael |author-last1=Luby |M.author-link1=Michael Luby]], |author-first2=M. |author-last2=Mitzenmacher, |author-first3=A. |author-last3=Shokrollahi, |author-first4=D. |author-last4=Spielman, |author-first5=V. |author-last5=Stemann |title=Practical Loss-Resilient Codes |journal=Proc. 29th Annual [[Association for Computing Machinery]] (ACM) Symposium on Theory of Computation |yeardate=1997}}</ref> Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word.<ref>{{Cite journal |title=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) |journal=En 302 307 |issue=V1.2.1 |publisher=[[ETSI]] |date=April 2009|postscript=<!--None-->}}</ref>

For turbo codes, an interleaver is an integral component and its proper design is crucial for good performance.<ref name="turbo-principles"/><ref>{{cite journal |author-first=K. |author-last=Andrews |title=The Development of Turbo and LDPC Codes for Deep-Space Applications |journal=[[Proc.Proceedings of the IEEE]] |volume=95 |issue=11 |date=November 2007 |display-authors=etal}}</ref> 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).<ref>S. Dolinar and D. Divsalar. Weight Distributions for Turbo Codes Using Random and Nonrandom Permutations. 1995. [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.105.6640&rep=rep1&type=pdf]</ref>

* Another possible construction is a contention-free quadratic [[permutation polynomial]] (QPP).<ref name="Takeshita1">{{cite journal | title=Permutation Polynomial Interleavers: An Algebraic-Geometric Perspective| year|date=2006 | author-first1=Oscar | author-last1= Takeshita | arxiv = cs/0601048 | postscript=<!--None--> | doi=10.1109/TIT.2007.896870 | volume=53 | issue=6 | journal=[[IEEE Transactions on Information Theory]] | pages=2116–2132}}</ref> It is used for example in the [[3GPP Long Term Evolution]] mobile telecommunication standard.<ref>[http://www.3gpp.org/ftp/Specs/html-info/36212.htm 3GPP TS 36.212], version 8.8.0, page 14</ref>

In multi-[[carrier signal|carrier]] communication systems, interleaving across carriers may be employed to provide frequency [[diversity scheme|diversity]], e.g., to mitigate [[frequency-selective fading]] or narrowband interference.<ref>{{Cite journal |title=Digital Video Broadcast (DVB); Frame structure, channel coding and modulation for a second generation digital terrestrial television broadcasting system (DVB-T2) |journal=En 302 755 |issue=V1.1.1 |publisher=[[ETSI]] |date=September 2009|postscript=<!--None-->}}</ref>

===Example===

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=== 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.<ref>{{cite web | title=Explaining Interleaving – W3techie | publisher=w3techie.com | url=http://w3techie.com/2010/explaining-interleaving/ | accessdateaccess-date=2010-06-03 }}</ref> 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|date=April 2017}}.

== Software for error-correcting codes ==

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* [[Repeat-accumulate code]]

* [[Repetition code]]s, such as [[Triple modular redundancy]]

* Spinal code, a rateless, nonlinear code based on pseudo-random hash functions <ref>{{Cite conference |author-first1=Jonathan |author-last1=Perry |author-first2=Hari |author-last2=Balakrishnan |author-first3=Devavrat |author-last3=Shah |title=Rateless Spinal Codes |booktitle=Proceedings of the 10th ACM Workshop on Hot Topics in Networks |date=2011 |url=http://doi.acm.org/10.1145/2070562.2070568 |doi=10.1145/2070562.2070568}}</ref>

first1=Jonathan | last1=Perry |

first2=Hari | last2=Balakrishnan |

first3=Devavrat | last3=Shah |

title = Rateless Spinal Codes |

booktitle = Proceedings of the 10th ACM Workshop on Hot Topics in Networks |

year = 2011 |

doi = 10.1145/2070562.2070568

}}</ref>

* [[Tornado code]], a near-optimal [[erasure code|erasure correcting code]], and the precursor to [[Fountain code]]s

* [[Turbo code]]

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==Further reading==

* {{cite book |author1author-last1=Clark, Jr. |author-first1=George C., Jr. |author2author-last2=Cain, |author-first2=J. Bibb |title=Error-Correction Coding for Digital Communications |publisher=[[Plenum Press]] |___location=New York, USA |yeardate=1981 |isbn=0-306-40615-2 |id={{ISBN|978-0-306-40615-7 }}}}

* {{cite book |author-last=Wicker, |author-first=Stephen B. |title=Error Control Systems for Digital Communication and Storage |publisher=[[Prentice-Hall]] |___location=Englewood Cliffs, NJ, USA |yeardate=1995 |isbn=0-13-200809-2 |id={{ISBN|978-0-13-200809-9 }}}}

* {{cite book |author-last=Wilson, |author-first=Stephen G. |title=Digital Modulation and Coding |publisher=[[Prentice-Hall]] |___location=Englewood Cliffs, NJ, USA |yeardate=1996 |isbn=0-13-210071-1 |id={{ISBN|978-0-13-210071-7 }}}}

* [https://web.archive.org/web/20070519161253/http://www.st.com/stonline/products/literature/an/10123.htm "Error Correction Code in Single Level Cell NAND Flash memories"] 16 February 2007

* [http://www.eetasia.com/ARTICLES/2004NOV/A/2004NOV29_MEM_AN10.PDF?SOURCES=DOWNLOAD "Error Correction Code in NAND Flash memories"] 29 November 2004

* [http://perspectives.mvdirona.com/2012/02/observations-on-errors-corrections-trust-of-dependent-systems/ Observations on Errors, Corrections, & Trust of Dependent Systems], by James Hamilton, 26 February 2012

* Sphere Packings, Lattices and Groups, By J. H. Conway, N. J. A. Sloane, [[Springer Science & Business Media]], 9 MarMarch 2013 – Mathematics – 682 pages.

==External links==

* {{Cite web |author-last=Morelos-Zaragoza |author-first=Robert |date=2004 |url=http://www.eccpage.com/ |title=The Correcting Codes (ECC) Page |access-date=2006-03-05}}

*{{Cite web

| last = Morelos-Zaragoza

| first = Robert

| year = 2004

| url = http://www.eccpage.com/

| title = The Correcting Codes (ECC) Page

| accessdate = 2006-03-05

}}

* [https://github.com/supermihi/lpdec lpdec: library for LP decoding and related things (Python)]