A New Family of MRD Codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb{F_q}^{2n\times2n}$ </tex-math> </inline-formula> With Right and Middle Nuclei <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{q^n}$ </tex-math> </inline-formula>

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.