New Semifields and new MRD Codes from Skew Polynomial Rings

Abstract

In this article we construct a new family of semifields, containing and extending two well-known families, namely Albert's generalised twisted fields and Petit's cyclic semifields (also known as Johnson-Jha semifields). The construction also gives examples of semifields with parameters for which no examples were previously known. In the case of semifields two dimensions over a nucleus and four-dimensional over their centre, the construction gives all possible examples. Furthermore we embed these semifields in a new family of maximum rank-distance codes, encompassing most known current constructions, including the (twisted) Delsarte-Gabidulin codes, and containing new examples for most parameters.

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Cited by in corpus (9)

• MRD codes with maximum idealizers
• Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants
• Nuclei and automorphism groups of generalized twisted Gabidulin codes
• MRD-codes arising from the trinomial $x^q+x^{q^3}+c{x}^{q^5}\in {\mathbb{F}}_{q^6}[x]$
• Connections between scattered linear sets and MRD-codes
• Asymptotics of Moore exponent sets
• Binary additive MRD codes with minimum distance n-1 must contain a semifield spread set
• On Matrix Algebras Isomorphic to Finite Fields and Planar Dembowski-Ostrom Monomials
• A geometric invariant of linear rank-metric codes