More MDS codes of non-Reed-Solomon type

TL;DR

This paper investigates a class of linear MDS codes distinct from Reed-Solomon codes, providing conditions for their MDS property, exploring near MDS cases, and analyzing their duals' covering radii and deep holes.

Contribution

It establishes new necessary and sufficient conditions for these codes to be MDS, expanding understanding beyond Reed-Solomon codes and identifying infinite families of extendable codes.

Findings

Identified conditions for MDS property in non-Reed-Solomon codes

Determined covering radii and deep holes of dual codes

Discovered infinite family of near MDS codes with dimension three

Abstract

MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and establish some sufficient and necessary conditions for them being MDS. Notably, these codes differ from Reed-Solomon codes up to monomial equivalence. Additionally, we also explore the cases in which these codes are almost MDS or near MDS. Applying our main results, we determine the covering radii and deep holes of the dual codes associated with specific Roth-Lempel codes and discover an infinite family of (almost) optimally extendable codes with dimension three.