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Reed–Solomon error correction
In information theory and coding theory, Reed–Solomon codes are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960. They have many applications, including consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, Data Matrix, data transmission technologies such as DSL and WiMAX, broadcast systems such as satellite communications, DVB and ATSC, and storage systems such as RAID 6.
Reed–Solomon codes operate on a block of data treated as a set of finite-field elements called symbols. Reed–Solomon codes RS(n, k) are able to detect and correct multiple symbol errors. By adding t = n − k check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to t erroneous symbols, or locate and correct up to ⌊t/2⌋ erroneous symbols at unknown locations. As an erasure code, it can correct up to 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 bit-error correcting codes, since a sequence of b + 1 consecutive bit errors can affect at most two symbols of size b. The choice of 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 – original view and BCH view – with BCH view being the most common, as BCH view decoders are faster and require less working storage than original view decoders.
Reed–Solomon codes were developed in 1960 by Irving S. Reed and Gustave Solomon, who were then staff members of MIT Lincoln Laboratory. Their seminal article was titled "Polynomial Codes over Certain Finite Fields". The original encoding scheme described in the Reed and Solomon article used a variable polynomial based on the message to be encoded where only a fixed set of values (evaluation points) to be encoded are known to encoder and decoder. The original theoretical decoder generated potential polynomials based on subsets of k (unencoded message length) out of n (encoded message length) values of a received message, choosing the most popular polynomial as the correct one, which was impractical for all but the simplest of cases. This was initially resolved by changing the original scheme to a BCH-code-like scheme based on a fixed polynomial known to both encoder and decoder, but later, practical decoders based on the original scheme were developed, although slower than the BCH schemes. The result of this is that there are two main types of Reed–Solomon codes: ones that use the original encoding scheme and ones that use the BCH encoding scheme.
Also in 1960, a practical fixed polynomial decoder for BCH codes developed by Daniel Gorenstein and Neal Zierler was described in an MIT Lincoln Laboratory report by Zierler in January 1960 and later in an article in June 1961. The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book "Error-Correcting Codes" by W. Wesley Peterson (1961).[page needed] By 1963 (or possibly earlier), J.J. Stone (and others)[who?] recognized that Reed–Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes, but Reed–Solomon codes based on the original encoding scheme are not a class of BCH codes, and depending on the set of evaluation points, they are not even cyclic codes.
In 1969, an improved BCH scheme decoder was developed by Elwyn Berlekamp and James Massey and has since been known as the Berlekamp–Massey decoding algorithm.
In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the extended Euclidean algorithm.
In 1977, Reed–Solomon codes were implemented in the Voyager program in the form of concatenated error correction codes. The first commercial application in mass-produced consumer products appeared in 1982 with the compact disc, where two interleaved Reed–Solomon codes are used. Today, Reed–Solomon codes are widely implemented in digital storage devices and digital communication standards, though they are being slowly replaced by Bose–Chaudhuri–Hocquenghem (BCH) codes. For example, Reed–Solomon codes are used in the Digital Video Broadcasting (DVB) standard DVB-S, in conjunction with a convolutional inner code, but BCH codes are used with LDPC in its successor, DVB-S2.
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Reed–Solomon error correction
In information theory and coding theory, Reed–Solomon codes are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960. They have many applications, including consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, Data Matrix, data transmission technologies such as DSL and WiMAX, broadcast systems such as satellite communications, DVB and ATSC, and storage systems such as RAID 6.
Reed–Solomon codes operate on a block of data treated as a set of finite-field elements called symbols. Reed–Solomon codes RS(n, k) are able to detect and correct multiple symbol errors. By adding t = n − k check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to t erroneous symbols, or locate and correct up to ⌊t/2⌋ erroneous symbols at unknown locations. As an erasure code, it can correct up to 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 bit-error correcting codes, since a sequence of b + 1 consecutive bit errors can affect at most two symbols of size b. The choice of 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 – original view and BCH view – with BCH view being the most common, as BCH view decoders are faster and require less working storage than original view decoders.
Reed–Solomon codes were developed in 1960 by Irving S. Reed and Gustave Solomon, who were then staff members of MIT Lincoln Laboratory. Their seminal article was titled "Polynomial Codes over Certain Finite Fields". The original encoding scheme described in the Reed and Solomon article used a variable polynomial based on the message to be encoded where only a fixed set of values (evaluation points) to be encoded are known to encoder and decoder. The original theoretical decoder generated potential polynomials based on subsets of k (unencoded message length) out of n (encoded message length) values of a received message, choosing the most popular polynomial as the correct one, which was impractical for all but the simplest of cases. This was initially resolved by changing the original scheme to a BCH-code-like scheme based on a fixed polynomial known to both encoder and decoder, but later, practical decoders based on the original scheme were developed, although slower than the BCH schemes. The result of this is that there are two main types of Reed–Solomon codes: ones that use the original encoding scheme and ones that use the BCH encoding scheme.
Also in 1960, a practical fixed polynomial decoder for BCH codes developed by Daniel Gorenstein and Neal Zierler was described in an MIT Lincoln Laboratory report by Zierler in January 1960 and later in an article in June 1961. The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book "Error-Correcting Codes" by W. Wesley Peterson (1961).[page needed] By 1963 (or possibly earlier), J.J. Stone (and others)[who?] recognized that Reed–Solomon codes could use the BCH scheme of using a fixed generator polynomial, making such codes a special class of BCH codes, but Reed–Solomon codes based on the original encoding scheme are not a class of BCH codes, and depending on the set of evaluation points, they are not even cyclic codes.
In 1969, an improved BCH scheme decoder was developed by Elwyn Berlekamp and James Massey and has since been known as the Berlekamp–Massey decoding algorithm.
In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the extended Euclidean algorithm.
In 1977, Reed–Solomon codes were implemented in the Voyager program in the form of concatenated error correction codes. The first commercial application in mass-produced consumer products appeared in 1982 with the compact disc, where two interleaved Reed–Solomon codes are used. Today, Reed–Solomon codes are widely implemented in digital storage devices and digital communication standards, though they are being slowly replaced by Bose–Chaudhuri–Hocquenghem (BCH) codes. For example, Reed–Solomon codes are used in the Digital Video Broadcasting (DVB) standard DVB-S, in conjunction with a convolutional inner code, but BCH codes are used with LDPC in its successor, DVB-S2.