A class of constacyclic codes are generalized Reed-Solomon codes

TL;DR

This paper investigates when MDS constacyclic codes are generalized Reed-Solomon codes by analyzing the conditions under which their squares have specific properties, providing new criteria for identifying GRS codes.

Contribution

It offers a sufficient condition and a necessary and sufficient condition for constacyclic codes to be GRS, especially focusing on prime length codes.

Findings

A study of the square of constacyclic codes.

A sufficient condition for constacyclic codes to be GRS.

A necessary and sufficient condition for prime length codes to be GRS.

Abstract

Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed-Solomon (GRS) codes. The square $\mathcal{C}^{2}$ of a linear code $\mathcal{C}$ is the linear code spanned by the component-wise products of every pair of codewords in $\mathcal{C}$. For an MDS code $\mathcal{C}$, it is convenient to determine whether $\mathcal{C}$ is a GRS code by determining the dimension of $\mathcal{C}^{2}$. In this paper, we investigate under what conditions that MDS constacyclic codes are GRS. For this purpose, we first study the square of constacyclic codes. Then, we give a sufficient condition that a constacyclic code is GRS. In particular, We provide a necessary and sufficient condition that a constacyclic code of a prime length is GRS.