A note on rank-metric codes

TL;DR

This paper characterizes the equivalence of a new class of maximum rank-metric codes introduced by Sheekey, explores their algebraic structures, and determines their automorphism groups, including those of twisted Gabidulin codes.

Contribution

It generalizes the known results on the equivalence and automorphisms of twisted Gabidulin codes to Sheekey's new class of MRD codes.

Findings

Characterization of code equivalence for Sheekey's codes

Identification of the right nucleus, middle nucleus, and duals of these codes

Calculation of the automorphism group and its size for these codes

Abstract

Let $\mathbb{F}_q$ denote the finite field with $q=p^\lambda$ elements. Maximum Rank-metric codes (MRD for short) are subsets of $M_{m\times n}(\mathbb{F}_q)$ whose number of elements attains the Singleton-like bound. The first MRD codes known was found by Delsarte (1978) and Gabidulin (1985). Sheekey (2016) presented a new class of MRD codes over $\mathbb{F}q$ called twisted Gabidulin codes and also proposed a generalization of the twisted Gabidulin codes to the codes $\mathcal{H}_{k,s}(L_1,L_2)$. The equivalence and duality of twisted Gabidulin codes was discussed by Lunardoni, Trombetti, and Zhou (2018). A new class of MRD codes in $M_{2n\times 2n}(\mathbb{F}_q)$ was found by Trombetti-Zhou (2018). In this work, we characterize the equivalence of the class of codes proposed by Sheekey, generalizing the results known for twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the paper, we restrict ourselves to the case $L_1(x)=x$, where we present its right nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section, we present the automorphism group of $\mathcal{H}_{k,s}(x,L(x))$ and compute its cardinality. In particular, we obtain the number of elements in the automorphism group of the twisted Gabidulin codes.