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In this paper, we present a new family of maximum rank-distance (MRD) codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{q}^{2n\times 2n}$ </tex-math></inline-formula> of minimum distance <inline-formula> <tex-math notation="LaTeX">$2\leq d\leq 2n$ </tex-math></inline-formula>. In particular, when <inline-formula> <tex-math notation="LaTeX">$d=2n$ </tex-math></inline-formula>, we can show that the corresponding semifield is exactly a Hughes&#x2013;Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{q^{n}}$ </tex-math></inline-formula>. We also prove that the MRD codes of minimum distance <inline-formula> <tex-math notation="LaTeX">$2&lt; d&lt; 2n$ </tex-math></inline-formula> in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.

ieee transactions on information theory

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>

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