Several classes of Galois self-orthogonal MDS codes and related applications

TL;DR

This paper introduces new classes of Galois self-orthogonal MDS codes and their applications, including quantum MDS codes, through novel construction methods for generalized Reed-Solomon codes.

Contribution

It proposes two general methods to construct $e$-Galois self-orthogonal GRS codes, leading to eight new classes and various applications in quantum coding.

Findings

Eight new classes of $e$-Galois self-orthogonal GRS codes for odd $q$ and $2e|h$.

Construction of new $e'$-Galois self-orthogonal MDS codes for all $e'$ in $[0,h-1]$.

Development of quantum MDS codes with lengths greater than $rac{ ext{sqrt}(q)}{+1}$ and high minimum distances.

Abstract

Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ and $h$ is even). In this paper, we propose two general methods to construct $e$-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of $e$-Galois self-orthogonal (extended) GRS codes with odd $q$ and $2e\mid h$ are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new $e'$-Galois self-orthogonal maximum distance separable (MDS) codes for all possible $e'$ satisfying $0\leq e'\leq h-1$, new $e$-Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional $e$-Galois hull via the punctured codes are derived. Moreover, some new $\sqrt{q}$-ary quantum MDS codes with lengths greater than $\sqrt{q}+1$ and minimum distances greater than $\frac{\sqrt{q}}{2}+1$ are obtained.