Alternant code

2020 Mathematics Subject Classification: Primary: 94Bxx [MSN][ZBL]

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 $H_{i,j} = \alpha_j^i y_i$, where the $\alpha_j$ 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 $\delta-1$, $j$ from 1 to $n$.

The parameters of this alternant code are length $n$, dimension $\ge n - m\delta$ and minimum distance $\ge \delta+1$. There exist long alternant codes which meet the Gilbert-Varshamov bound.

The class of alternant codes includes BCH codes, Goppa codes and Srivastava codes.

References

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