Automorphism groups of Gabidulin-like codes

TL;DR

This paper characterizes the automorphism groups of Gabidulin-like codes, a class of maximum rank distance codes constructed over cyclic Galois extensions, unifying existing constructions.

Contribution

It provides a complete characterization of the automorphism groups of Gabidulin-like codes and identifies the conditions under which codes are equivalent to these constructions.

Findings

Automorphism groups are explicitly determined for Gabidulin-like codes.

The construction unifies various existing rank-metric code families.

Conditions for code equivalence to Gabidulin-like codes are established.

Abstract

Let K be a cyclic Galois extension of degree f over k and T a generator of the Galois group. For any v=(v_1,... , v_m)\in K^m such that v is linearly independent over k, and any 0< d < m the Gabidulin-like code C(v, T , d) is a maximum rank distance code in the space of f times m matrices over k of dimension fd. This construction unifies the ones available in the literature. We characterise the K-linear codes that are Gabidulin-like codes and determine their rank-metric automorphism group.