The equivalence of GRS codes and EGRS codes

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.