Alternant code
Encyclopedia
In coding theory
, alternant codes form a class of parameterised error-correcting codes
which generalise the BCH code
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form Hi,j = αjiyi, where the αj are distinct elements of the extension GF(qm), the yi are further non-zero parameters again in the extension GF(qm) and the indices range as i from 0 to δ − 1, j from 1 to n.
There exist long alternant codes which meet the Gilbert-Varshamov bound
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The class of alternant codes includes
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
, alternant codes form a class of parameterised error-correcting codes
Error detection and correction
In information theory and coding theory with applications in computer science and telecommunication, error detection and correction or error control are techniques that enable reliable delivery of digital data over unreliable communication channels...
which generalise the BCH code
BCH code
In coding theory the BCH codes form a class of parameterised error-correcting codes which have been the subject of much academic attention in the last fifty years. BCH codes were invented in 1959 by Hocquenghem, and independently in 1960 by Bose and Ray-Chaudhuri...
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Definition
An alternant code over GF(q) of length n is defined by a parity check matrix H of alternantAlternant matrix
In linear algebra, an alternant matrix, is a matrix with a particular structure, in which successive columns have a particular function applied to their entries. An alternant determinant is the determinant of an alternant matrix...
form Hi,j = αjiyi, where the αj are distinct elements of the extension GF(qm), the yi are further non-zero parameters again in the extension GF(qm) and the indices range as i from 0 to δ − 1, j from 1 to n.
Properties
The parameters of this alternant code are length n, dimension ≥ n − mδ and minimum distance ≥ δ + 1.There exist long alternant codes which meet the Gilbert-Varshamov bound
Gilbert-Varshamov bound
In coding theory, the Gilbert–Varshamov bound is a bound on the parameters of a code . It is occasionally known as the Gilbert–Shannon–Varshamov bound , but the Gilbert–Varshamov bound is by far the most popular name...
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The class of alternant codes includes
- BCH codeBCH codeIn coding theory the BCH codes form a class of parameterised error-correcting codes which have been the subject of much academic attention in the last fifty years. BCH codes were invented in 1959 by Hocquenghem, and independently in 1960 by Bose and Ray-Chaudhuri...
s - Goppa codes
- Srivastava codeSrivastava codeIn coding theory, Srivastava codes form a class of parameterised error-correcting codes which are a special case of alternant codes.-Definition:...
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