A skew polynomial framework for constructing division algebras and linear maximum rank distance codes

We construct division algebras and linear maximum rank distance (MRD) matrix codes using skew polynomials over fields. The non-unital division algebras we obtain generalize several prominent constructions: Sheekey’s twisted cyclic pre-semifields, i.e. the pre-semifields associated with Jha-Johnson semifields and the semifields associated with Albert’s generalized twisted fields. Our linear MRD codes generalize the constructions of Lobillo, Santonastaso and Sheekey. We present criteria for these algebras to be division algebras, respectively, for when these codes have maximum rank, and compare isotopic division algebras that appear throughout recent and classical literature. We compute some of their invariants.