New MDS Self-Dual Codes from Generalized Reed-Solomon Codes

TL;DR

This paper constructs new classes of MDS self-dual codes over odd finite fields using generalized Reed-Solomon codes, expanding known existence results for various lengths and field sizes.

Contribution

It introduces novel constructions of MDS self-dual codes over odd fields, particularly for even lengths and specific field sizes, filling gaps in existing literature.

Findings

Existence of q-ary MDS self-dual codes for even length n when q ≡ 1 mod 4 and q is large.

Construction of MDS self-dual codes over q = r^2 with conditions on n, including n ≤ r and divisibility properties.

Identification of new parameter sets where MDS self-dual codes exist over odd finite fields.

Abstract

Both MDS and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining existence of $q$-ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where $q$ is even. The current paper focuses on the case where $q$ is odd. We construct a few classes of new MDS self-dual code through generalized Reed-Solomon codes. More precisely, we show that for any given even length $n$ we have a $q$-ary MDS code as long as $q\equiv1\bmod{4}$ and $q$ is sufficiently large (say $q\ge 2^n\times n^2)$. Furthermore, we prove that there exists a $q$-ary MDS self-dual code of length $n$ if $q=r^2$ and $n$ satisfies one of the three conditions: (i) $n\le r$ and $n$ is even; (ii) $q$ is odd and $n-1$ is an odd divisor of $q-1$; (iii) $r\equiv3\mod{4}$ and $n=2tr$ for any $t\le (r-1)/2$.