Generalized Twisted Gabidulin Codes

TL;DR

This paper introduces a new family of maximum rank distance (MRD) codes called generalized twisted Gabidulin codes, analyzes their properties, and determines the conditions under which different codes in this family are equivalent.

Contribution

It generalizes existing Gabidulin and twisted Gabidulin codes, computes their duals and adjoints, and fully characterizes their equivalence classes.

Findings

Generalized twisted Gabidulin codes include known MRD codes as subsets.

The paper computes duals and adjoints of these codes.

It establishes criteria for code equivalence within the family.

Abstract

Let $\mathcal{C}$ be a set of $m$ by $n$ matrices over $\mathbb{F}_q$ such that the rank of $A-B$ is at least $d$ for all distinct $A,B\in \mathcal{C}$. Suppose that $m\leqslant n$. If $\#\mathcal{C}= q^{n(m-d+1)}$, then $\mathcal{C}$ is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary $1<d<m-1$. One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was later generalized by Kshevetskiy and Gabidulin (2005). We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016), and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.