A New Family of MRD Codes in $\mathbb{F_q}^{2n\times2n}$ With Right and Middle Nuclei $\mathbb F_{q^n}$.

In this paper, we present a new family of maximum rank-distance (MRD) 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. [ABSTRACT FROM AUTHOR]