The equivalence of GRS codes and EGRS codes

TL;DR

This paper proves that over finite fields, GRS codes and EGRS codes are essentially the same, establishing their equivalence in the context of linear MDS codes.

Contribution

It demonstrates that any linear code over a finite field that is GRS is also EGRS, and vice versa, clarifying their relationship in coding theory.

Findings

GRS codes are equivalent to EGRS codes over finite fields.

The proof applies to codes with length between 2 and q.

This result simplifies understanding of MDS codes in coding theory.

Abstract

Generalized Reed-Solomon and extended generalized Reed-Solomon (abbreviation to GRS and EGRS) codes are the most well-known family of MDS codes with wide applications in coding theory and practice. Let $\mathbb{F}_q$ be the $q$ elements finite field, where $q$ is the power of a prime. For a linear code $\mathcal{C}$ over $\mathbb{F}_q$ with length $2\le n\le q$, we prove that $\mathcal{C}$ is a GRS code if and only if $\mathcal{C}$ is a EGRS code.