Division algebras and MRD codes from skew polynomials

TL;DR

This paper constructs division algebras and generalized MRD codes from skew polynomial rings over division algebras, extending known classes like Gabidulin codes and Jha Johnson semifields.

Contribution

It introduces a new framework for creating division algebras and MRD codes using skew polynomials over noncommutative division algebras.

Findings

Construction of new division algebras from skew polynomials.

Generalization of MRD codes including known classes.

Connection between algebraic structures and coding theory.

Abstract

Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\sigma,\delta]$ a skew polynomial ring. Using skew polynomials $f\in R$, we construct division algebras and a generalization of maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include a class of codes constructed by Sheekey (in particular, generalized Gabidulin codes), as well as Jha Johnson semifields.