New Semifields and new MRD Codes from Skew Polynomial Rings

TL;DR

This paper introduces a new family of semifields derived from skew polynomial rings, extending known classes and providing new examples, which are then used to construct a broad family of maximum rank-distance codes with improved parameters.

Contribution

The authors construct new semifields from skew polynomial rings that extend existing families and generate a comprehensive family of MRD codes, including many previously unknown parameter sets.

Findings

Constructed new semifields extending Albert's and Petit's families.

Provided all possible semifields with specific parameters.

Developed a new family of MRD codes encompassing most known constructions.

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.