Nuclei and automorphism groups of generalized twisted Gabidulin codes

TL;DR

This paper investigates the automorphism groups of generalized twisted Gabidulin codes for the case where the matrix dimensions are not equal, focusing on nuclei invariants to understand their structure.

Contribution

It extends the understanding of automorphism groups of generalized twisted Gabidulin codes to the case m<n by analyzing nuclei invariants and deriving necessary conditions.

Findings

Determined middle and right nuclei under certain conditions

Derived necessary conditions for automorphisms

Extended automorphism group analysis to m<n case

Abstract

Generalized twisted Gabidulin codes are one of the few known families of maximum rank matrix codes over finite fields. As a subset of m by n matrices, when m=n, the automorphism group of any generalized twisted Gabidulin code has been completely determined recently. In this paper, we consider the same problem for m<n. Under certain conditions on their parameters, we determine their middle nuclei and right nuclei, which are important invariants with respect to the equivalence for rank metric codes. Furthermore, we also use them to derive necessary conditions on the automorphisms of generalized twisted Gabidulin codes.