Editing Reed–Solomon error correction (section)

== Applications ==

===Data storage===
Reed–Solomon coding is very widely used in mass storage systems to correct
the burst errors associated with media defects.

Reed–Solomon coding is a key component of the [[compact disc]]. It was the first use of strong error correction coding in a mass-produced consumer product, and [[digital audio tape|DAT]] and [[DVD]] use similar schemes. In the CD, two layers of Reed–Solomon coding separated by a 28-way [[convolution]]al [[interleaver]] yields a scheme called Cross-Interleaved Reed–Solomon Coding ([[Cross-interleaved Reed–Solomon coding|CIRC]]). The first element of a CIRC decoder is a relatively weak inner (32,28) Reed–Solomon code, shortened from a (255,251) code with 8-bit symbols. This code can correct up to 2 byte errors per 32-byte block. More importantly, it flags as erasures any uncorrectable blocks, i.e., blocks with more than 2 byte errors. The decoded 28-byte blocks, with erasure indications, are then spread by the deinterleaver to different blocks of the (28,24) outer code. Thanks to the deinterleaving, an erased 28-byte block from the inner code becomes a single erased byte in each of 28 outer code blocks. The outer code easily corrects this, since it can handle up to 4 such erasures per block.

The result is a CIRC that can completely correct error bursts up to 4000 bits, or about 2.5&nbsp;mm on the disc surface. This code is so strong that most CD playback errors are almost certainly caused by tracking errors that cause the laser to jump track, not by uncorrectable error bursts.<ref>{{Citation |first=K. A. S. |last=Immink |author-link=Kees Immink |contribution=Reed–Solomon Codes and the Compact Disc |editor1-first=Stephen B. |editor1-last=Wicker |editor2-first=Vijay K. |editor2-last=Bhargava |title=Reed–Solomon Codes and Their Applications |publisher=[[IEEE Press]] |year=1994 |isbn=978-0-7803-1025-4 }}</ref>

DVDs use a similar scheme, but with much larger blocks, a (208,192) inner code, and a (182,172) outer code.

Reed–Solomon error correction is also used in [[parchive]] files which are commonly posted accompanying multimedia files on [[USENET]]. The distributed online storage service [[Wuala]] (discontinued in 2015) also used Reed–Solomon when breaking up files.

===Bar code===
Almost all two-dimensional bar codes such as [[PDF-417]], [[MaxiCode]], [[Datamatrix]], [[QR Code]], [[Aztec Code]] and [[Han Xin code]] use Reed–Solomon error correction to allow correct reading even if a portion of the bar code is damaged. When the bar code scanner cannot recognize a bar code symbol, it will treat it as an erasure.

Reed–Solomon coding is less common in one-dimensional bar codes, but is used by the [[PostBar]] symbology.

===Data transmission===
Specialized forms of Reed–Solomon codes, specifically [[Cauchy matrix|Cauchy]]-RS and [[Vandermonde matrix|Vandermonde]]-RS, can be used to overcome the unreliable nature of data transmission over [[Binary erasure channel|erasure channels]]. The encoding process assumes a code of RS(''N'',&nbsp;''K'') which results in ''N'' codewords of length ''N'' symbols each storing ''K'' symbols of data, being generated, that are then sent over an erasure channel.

Any combination of ''K'' codewords received at the other end is enough to reconstruct all of the ''N'' codewords. The code rate is generally set to 1/2 unless the channel's erasure likelihood can be adequately modelled and is seen to be less. In conclusion, ''N'' is usually 2''K'', meaning that at least half of all the codewords sent must be received in order to reconstruct all of the codewords sent.

Reed–Solomon codes are also used in [[xDSL]] systems and [[CCSDS]]'s [[Space Communications Protocol Specifications]] as a form of [[forward error correction]].

===Space transmission===

[[File:DeepSpaceFEC.png|350px|right|thumb| Deep-space concatenated coding system.<ref>{{cite book |first1=J. |last1=Hagenauer |first2=E. |last2=Offer |first3=L. |last3=Papke |chapter=11. Matching Viterbi Decoders and Reed-Solomon Decoders in a Concatenated System |title=Reed Solomon Codes and Their Applications |publisher=IEEE Press |date=1994 |page=433 |oclc=557445046 |isbn=9780470546345}}</ref> Notation: RS(255, 223) + [[convolutional codes|CC]] ("constraint length" = 7, code rate = 1/2).]]

One significant application of Reed–Solomon coding was to encode the digital pictures sent back by the [[Voyager program]].

Voyager introduced Reed–Solomon coding [[concatenated code|concatenated]] with [[convolutional code]]s, a practice that has since become very widespread in deep space and satellite (e.g., direct digital broadcasting) communications.
<!-- Unsourced image removed: [[Image:NASA ECC Codes-imperfection.png|thumb|600px|none|NASA's Deep Space Missions ECC Codes (code imperfectness) {{Deletable image-caption|date=March 2012}}]] -->

[[Viterbi decoder]]s tend to produce errors in short bursts. Correcting these burst errors is a job best done by short or simplified Reed–Solomon codes.

Modern versions of concatenated Reed–Solomon/Viterbi-decoded convolutional coding were and are used on the [[Mars Pathfinder]], [[Galileo probe|Galileo]], [[Mars Exploration Rover]] and [[Cassini probe|Cassini]] missions, where they perform within about 1–1.5 [[decibel|dB]] of the ultimate limit, the [[Channel capacity|Shannon capacity]].

These concatenated codes are now being replaced by more powerful [[turbo code]]s:
{| class="wikitable"
|+ Channel coding schemes used by NASA missions<ref name="Andrews, 2007">{{cite journal |last1=Andrews |first1=K.S. |last2=Divsalar |first2=D. |last3=Dolinar |first3=S. |last4=Hamkins |first4=J. |last5=Jones |first5=C.R. |last6=Pollara |first6=F. |title=The development of turbo and LDPC codes for deep-space applications. |journal=Proceedings of the IEEE |volume=95 |issue=11 |pages=2142–56 |date=2007 |doi=10.1109/JPROC.2007.905132 |s2cid=9289140 |url=https://scholar.archive.org/work/shkuo6oxabbklkfz4d6v4gero4/access/wayback/http://coding.jpl.nasa.gov/~hamkins/publications/journals/2007_11_turbo_LDPC.pdf}}</ref>
|-
! Years  !! Code !! Mission(s)
|-
| 1958–present || Uncoded || Explorer, Mariner, many others
|-
| 1968–1978 || [[convolutional codes]] (CC) (25, 1/2) || Pioneer, Venus
|-
| 1969–1975 || [[Reed–Muller code]] (32, 6) || Mariner, Viking
|-
| 1977–present || [[Binary Golay code]] || Voyager
|-
| 1977–present || RS(255, 223) + CC(7, 1/2) || Voyager, Galileo, many others
|-
| 1989–2003 || RS(255, 223) + CC(7, 1/3) || Voyager
|-
| 1989–2003 || RS(255, 223) + CC(14, 1/4) || Galileo
|-
| 1996–present || RS + CC (15, 1/6) || Cassini, Mars Pathfinder, others	
|-
| 2004–present || [[Turbo codes]]{{refn|group=nb| Authors in Andrews et al. (2007), provide simulation results which show that for the same code rate (1/6) turbo codes outperform Reed-Solomon concatenated codes up to 2 dB ([[bit error rate]]).<ref name="Andrews, 2007"/>}}
|| Messenger, Stereo, MRO, MSL,others
|-
| est. 2009 || [[LDPC codes]] || Constellation, M2020, MAVEN
|}