# On the number of inequivalent Gabidulin codes

## TL;DR

This paper investigates the diversity of Gabidulin codes within the class of maximum rank-distance codes, revealing a large set of inequivalent codes for certain matrix dimensions.

## Contribution

It provides a detailed analysis of the equivalence problem for Gabidulin codes, showing many are pairwise inequivalent when 2 ≤ m ≤ n-2.

## Key findings

- Large subset of pairwise inequivalent Gabidulin codes identified
- Equivalence problem for Gabidulin codes analyzed in detail
- Results applicable for matrix dimensions where 2 ≤ m ≤ n-2

## Abstract

Maximum rank-distance (MRD) codes are extremal codes in the space of $m\times n$ matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are today known as Gabidulin codes. Motivated by several recent approaches to construct MRD codes that are inequivalent to Gabidulin codes, we study the equivalence issue for Gabidulin codes themselves. This shows in particular that the family of Gabidulin codes already contains a huge subset of MRD codes that are pairwise inequivalent, provided that $2\le m\le n-2$.