Editing Reed–Solomon error correction (section)

{{short description|Error-correcting codes}}
{{infobox code
 | name           = Reed–Solomon codes
 | image          =
 | image_caption  =
 | namesake       = [[Irving S. Reed]] and [[Gustave Solomon]]
 | hierarchy      = [[Linear block code]]<br />[[Polynomial code]]<br />Reed–Solomon code
 | block_length   = ''n''
 | message_length = ''k''
 | distance       = ''n'' − ''k'' + 1
 | alphabet_size  = ''q'' = ''p''<sup>''m''</sup> ≥ ''n''&nbsp; (''p'' prime)<br />Often ''n'' = ''q'' − 1.
 | notation       = [''n'', ''k'', ''n'' − ''k'' + 1]<sub>''q''</sub>-code
 | decoding       = [[Berlekamp–Massey algorithm|Berlekamp–Massey]]<br />[[Euclidean algorithm|Euclidean]]<br />''et al.''
 | properties     = [[maximum distance separable code|Maximum-distance separable code]]
}}
In [[information theory]] and [[coding theory]], '''Reed–Solomon codes''' are a group of [[error-correcting code]]s that were introduced by [[Irving S. Reed]] and [[Gustave Solomon]] in 1960.<ref name="ReedSolomon">{{cite journal |last1=Reed |first1= Irving S. |author-link=Irving S. Reed |last2=Solomon |first2=Gustave |author-link2=Gustave Solomon |title=Polynomial Codes over Certain Finite Fields |journal=Journal of the Society for Industrial and Applied Mathematics |volume=8 |issue=2 |pages=300–304 |year= 1960 |url= https://sites.math.rutgers.edu/~zeilberg/akherim/ReedS1960.pdf |doi=10.1137/0108018}}</ref>
They have many applications, including consumer technologies such as [[MiniDisc]]s, [[CD]]s, [[DVD]]s, [[Blu-ray]] discs, [[QR code]]s, [[Data Matrix]], [[data transmission]] technologies such as [[DSL]] and [[WiMAX]], [[Broadcasting|broadcast]] systems such as satellite communications, [[Digital Video Broadcasting|DVB]] and [[ATSC Standards|ATSC]], and storage systems such as [[RAID 6]].

Reed–Solomon codes operate on a block of data treated as a set of [[finite field|finite-field]] elements called symbols. Reed–Solomon codes are able to detect and correct multiple symbol errors. By adding {{math|''t'' {{=}} ''n'' − ''k''}} check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to {{mvar|t}} erroneous symbols, ''or'' locate and correct up to {{math|⌊''t''/2⌋}} erroneous symbols at unknown locations. As an [[erasure code]], it can correct up to {{mvar|t}} erasures at locations that are known and provided to the algorithm, or it can detect and correct combinations of errors and erasures. Reed–Solomon codes are also suitable as multiple-[[burst error|burst]] bit-error correcting codes, since a sequence of {{math|''b''&nbsp;+&nbsp;1}} consecutive bit errors can affect at most two symbols of size {{mvar|b}}. The choice of {{mvar|t}} is up to the designer of the code and may be selected within wide limits.

There are two basic types of Reed–Solomon codes{{snd}} original view and [[BCH code|BCH]] view{{snd}} with BCH view being the most common, as BCH view decoders are faster and require less working storage than original view decoders.