Galois self-orthogonal MDS codes with large dimensions

TL;DR

This paper characterizes conditions for $e$-Galois self-orthogonality in generalized Reed-Solomon codes, constructs new classes of MDS codes with large dimensions, and explores applications to quantum code construction.

Contribution

It provides necessary and sufficient conditions for $e$-Galois self-orthogonality in GRS codes and introduces new MDS codes with large dimensions and arbitrary Galois hulls.

Findings

Classified $e$-Galois self-orthogonal codes into three cases.

Constructed new $e$-Galois self-orthogonal MDS codes with larger dimensions.

Generated quantum codes from the new MDS codes with Galois hulls.

Abstract

Let $q=p^m$ be a prime power, $e$ be an integer with $0\leq e\leq m-1$ and $s=\gcd(e,m)$. In this paper, for a vector $v$ and a $q$-ary linear code $C$, we give some necessary and sufficient conditions for the equivalent code $vC$ of $C$ and the extended code of $vC$ to be $e$-Galois self-orthogonal. From this, we directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be $e$-Galois self-orthogonal. Furthermore, for all possible $e$ satisfying $0\leq e\leq m-1$, we classify them into three cases (1) $\frac{m}{s}$ odd and $p$ even; (2) $\frac{m}{s}$ odd and $p$ odd; (3) $\frac{m}{s}$ even, and construct several new classes of $e$-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our $e$-Galois self-orthogonal MDS codes can have dimensions greater than $\lfloor \frac{n+p^e-1}{p^e+1}\rfloor$, which are not covered by previously known ones. Moreover, by propagation rules, we obtain some new MDS codes with Galois hulls of arbitrary dimensions. As an application, many quantum codes can be obtained from these MDS codes with Galois hulls.