The equivalence of linear codes implies semi-linear equivalence

TL;DR

This paper proves that linear code equivalence implies semi-linear equivalence, and additive MDS code equivalence implies additive equivalence, establishing fundamental relationships between code equivalences.

Contribution

It establishes that linear code equivalence necessarily implies semi-linear equivalence and additive MDS code equivalence implies additive equivalence, clarifying the structure of code equivalences.

Findings

Linear code equivalence implies semi-linear equivalence

Additive MDS code equivalence implies additive equivalence

Fundamental relationships between different code equivalences

Abstract

We prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent then they are additively equivalent.