Alternant code

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In coding theory, alternant codes form a class of parameterised error-correcting codes which generalise the BCH codes.

An alternant code over GF(q) of length n is defined by a parity check matrix H of alternant form H**i,j = αjiy**i, where the αj are distinct elements of the extension GF(q**m), the y**i are further non-zero parameters again in the extension GF(q**m) and the indices range as i from 0 to δ − 1, j from 1 to n.

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.

The class of alternant codes includes

• BCH codes
• Goppa codes
• Srivastava codes

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