A new family of MRD codes in ${\mathbb{F}}_q^{2n\times 2n}$ with right and middle nuclei ${\mathbb{F}}_{q^n}$

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 $2\le d\le 2n$. 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.

Compared with edition 1, several typos are corrected

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Cited by in corpus (9)

• Generalized Twisted Gabidulin Codes
• A Characterization of the Number of Roots of Linearized and Projective Polynomials in the Field of Coefficients
• New Semifields and new MRD Codes from Skew Polynomial Rings
• MRD codes with maximum idealizers
• Identifiers for MRD-codes
• Nuclei and automorphism groups of generalized twisted Gabidulin codes
• Automorphism groups and new constructions of maximum additive rank metric codes with restrictions
• A note on rank-metric codes
• Binary additive MRD codes with minimum distance n-1 must contain a semifield spread set