Invariants and Inequivalence of Linear Rank-Metric Codes

TL;DR

This paper introduces invariants based on field automorphisms for linear rank-metric codes, providing a new criterion to determine code inequivalence and deriving bounds on the number of code classes.

Contribution

It presents a novel invariant for linear rank-metric codes and uses it to establish a criterion for inequivalence and bounds on code class counts.

Findings

Invariant based on automorphisms distinguishes code classes

Criterion efficiently checks code inequivalence

Bounds on the number of code equivalence classes derived

Abstract

We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, is an invariant for the whole equivalence class of the code. These invariants give rise to an easily computable criterion to check if two codes are inequivalent. With this criterion we then derive bounds on the number of equivalence classes of classical and twisted Gabidulin codes.