Difference between revisions of "Alternant code"
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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 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 code]]s, [[Goppa code]]s and [[ | + | The class of alternant codes includes [[BCH code]]s, [[Goppa code]]s and [[Srivastava code]]s. |
== References == | == References == | ||
| − | * F.J. MacWilliams, N.J.A. Sloane. | + | * F.J. MacWilliams, N.J.A. Sloane. ''The Theory of Error-Correcting Codes'' (North-Holland, 1977) {{ISBN|0-444-85193-3}}, pp.332-338 |
Latest revision as of 20:50, 5 December 2023
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
Alternant code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alternant_code&oldid=30393