Construction of non-generalized Reed-Solomon MDS codes based on systematic generator matrix

TL;DR

This paper constructs non-GRS MDS codes using systematic generator matrices, expanding the class of known optimal codes beyond generalized Reed-Solomon codes.

Contribution

It proves two families of GTRS codes are non-GRS and provides a systematic generator matrix for a class of GTRS codes, introducing a new construction method for non-GRS MDS codes.

Findings

Two families of GTRS codes are non-GRS.

A systematic generator matrix for GTRS codes is provided.

A new construction method for non-GRS MDS codes is proposed.

Abstract

Maximum distance separable (MDS) codes are considered optimal because the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are likely the generalized Reed-Solomon (GRS) codes. In 1989, Roth and Lempel constructed a type of MDS code that is not a GRS code (referred to as non-GRS). In 2017, Beelen et al. introduced twisted Reed-Solomon (TRS) codes and demonstrated that many MDS TRS codes are indeed non-GRS. Following this, the definition of TRS codes was generalized to the most comprehensive form, which we refer to as generalized twisted Reed-Solomon (GTRS) codes. In this paper, we prove that two families of GTRS codes are non-GRS and provide a systematic generator matrix for a class of GTRS codes. Inspired by the form of the systematic generator matrix for GTRS codes,we also present a construction of non-GRS MDS codes.