# 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

**How to Cite This Entry:**

Alternant code.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Alternant_code&oldid=54744