For example, Reed–Solomon codes are used in the Digital Video Broadcasting (DVB) standard DVB-S, but LDPC codes are used in its successor, DVB-S2. Today, Reed–Solomon codes are widely implemented in digital storage devices and digital communication standards, though they are being slowly replaced by more modern low-density parity-check (LDPC) codes or turbo 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. In 1977, Reed–Solomon codes were implemented in the Voyager program in the form of concatenated error correction codes. In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the extended Euclidean algorithm.
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.
#Solomon word generator#
Stone (and others) 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. The Gorenstein–Zierler decoder and the related work on BCH codes are described in a book Error Correcting Codes by W. 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.Īlso 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 a paper in June 1961.
#Solomon word code#
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 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. The original encoding scheme described in the Reed & 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. Their seminal article was titled "Polynomial Codes over Certain Finite Fields". Reed and Gustave Solomon, who were then staff members of MIT Lincoln Laboratory. Reed–Solomon codes were developed in 1960 by Irving S.
5.5 Decoding beyond the error-correction bound.5.4 Decoder using discrete Fourier transform.5.1.5 Find the roots of the error locator polynomial.5.1 Peterson–Gorenstein–Zierler decoder.3.2 The BCH view: The codeword as a sequence of coefficients.3.1.3 Discrete Fourier transform and its inverse.3.1.2 Systematic encoding procedure: The message as an initial sequence of values.