Alternant code

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.

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