A new family of MRD codes in $\mathbb F_q^{2n\times2n}$ with right and middle nuclei $\mathbb F_{q^n}$

TL;DR

This paper introduces a new family of maximum rank distance (MRD) codes in matrix spaces over finite fields, revealing their nuclei structure and inequivalence to known codes, with connections to semifields.

Contribution

It presents a novel family of MRD codes with specific nuclei properties and proves their inequivalence to existing codes, expanding the landscape of rank-metric code constructions.

Findings

MRD codes with nuclei $_{q^n}$ are constructed.

When $d=2n$, codes correspond to Hughes-Kleinfeld semifields.

Codes with $2<d<2n$ are inequivalent to known codes.

Abstract

In this paper, we present a new family of maximum rank distance (MRD for short) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$. In particular, when $d=2n$, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^n}$. We also prove that the MRD codes of minimum distance $2<d<2n$ in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.