Division algebras and MRD codes from skew polynomials

Abstract

Let D be a division algebra, finite-dimensional over its center, and R = D[t; σ, δ] a skew polynomial ring.Using skew polynomials f ∈ 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.