Error detection and correction

Source: Wikipedia. Pages: 95. Chapters: Hash function, Reed-Solomon error correction, Hamming code, Longitudinal redundancy check, Forward error correction, BCH code, Sanity testing, Pearson hashing, Latin square, Convolutional code, Automatic repeat request, Hagelbarger code, Negative-acknowledge character, Summation check, Transverse redundancy check, Pseudo bit error ratio, Forward-backward algorithm, Hamming(7,4), List-decoding, Viterbi algorithm, Coding theory, Low-density parity-check code, Turbo code, Reed-Muller code, Check digit, Viterbi decoder, Snake-in-the-box, Verhoeff algorithm, Parity bit, Interleaving, Casting out nines, Hybrid automatic repeat request, Group code recording, Luhn algorithm, Concatenated error correction code, Berlekamp-Massey algorithm, Binary Golay code, Hash tree, Triple-channel architecture, Selective Repeat ARQ, Sequential decoding, Coding gain, Repetition code, Constant-weight code, Triple modular redundancy, Maximum likelihood sequence estimation, Hadamard code, Automated Quality control of meteorological observations, Go-Back-N ARQ, Berger code, Chien search, Link adaptation, Stop-and-wait ARQ, Enumerator polynomial, Hash list, Error-correcting codes with feedback, Shaping codes, Remote Error Indication, EXIT chart, Redundant array of independent memory, Locally decodable code, Walsh code, Cross-interleaved Reed-Solomon coding, Chipkill, Iterative Viterbi decoding, BCJR algorithm, Residual bit error rate, Justesen code, Preparata code, Locally testable code, Header checksum, Lexicographic code, Multidimensional parity-check code, Soft output Viterbi algorithm, Majority logic decoding, Bipolar violation, Alternant code, Coset leader, Repeat-accumulate code, Memory ProteXion, Srivastava code, Error correction model, Header Check Sequence, Detection Error Tradeoff, Long code, Soft-decision decoder, Data scrubbing, Sparse graph code, Hardened Core, Time triple modular redundancy, Soft-in soft-out decoder, Zigzag code. Excerpt: In coding theory, Reed-Solomon (RS) codes are non-binary cyclic error-correcting codes invented by Irving S. Reed and Gustave Solomon. They described a systematic way of building codes that could detect and correct multiple random symbol errors. By adding t check symbols to the data, an RS code can detect any combination of up to t erroneous symbols, and correct up to ¿t/2¿ symbols. As an erasure code, it can correct up to t known erasures, or it can detect and correct combinations of errors and erasures. Furthermore, RS codes are 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. In Reed-Solomon coding, source symbols are viewed as coefficients of a polynomial p(x) over a finite field. The original idea was to create n code symbols from k source symbols by oversampling p(x) at n > k distinct points, transmit the sampled points, and use interpolation techniques at the receiver to recover the original message. That is not how RS codes are used today. Instead, RS codes are viewed as cyclic BCH codes, where encoding symbols are derived from the coefficients of a polynomial constructed by multiplying p(x) with a cyclic generator polynomial. This gives rise to an efficient decoding algorithm, which was discovered by Elwyn Berlekamp and James Massey, and is known as the Berlekamp-Massey decoding algor...