Alternant code

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

Definition

An alternant code over GF(q) of length n is defined by a parity check matrix H of alternant form H_i,j = α_jⁱy_i, where the α_j 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.

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.

The class of alternant codes includes

• BCH codes
• Goppa codes
• Srivastava codes

References

• F.J. MacWilliams; N.J.A. Sloane (1977). The Theory of Error-Correcting Codes. North-Holland. pp. 332–338. ISBN 0-444-85193-3.