Division algebras and MRD codes from skew polynomials

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 maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include Jha Johnson semifields, and the classes of classical and twisted Gabidulin codes constructed by Sheekey.