Quotients of skew polynomial rings: new constructions of division algebras and MRD codes
F. J. Lobillo, Paolo Santonastaso, John Sheekey
TL;DR
The paper develops a unified skew-polynomial framework to construct (not necessarily associative) division algebras and MRD codes via quotients $R/{RF(x^n)}$ of the Ore extension $R={\mathbb L}[x;\sigma]$. It extends previous results to arbitrary $\ell_F=n/m$ by explicitly handling cases with $1<\ell_F<n$ and by analyzing bounds, eigenrings, and nucleus/idealiser structures, both over finite and infinite division rings. Two new families of MRD codes and division algebras are introduced: (i) a first family $S_{n,s\ell,k}(\eta,\rho,F)$ extending earlier finite-field constructions to general $\ell_F$, and (ii) a second family $D_{n,s\ell,k}(\gamma,F)$ extending Hughes–Kleinfeld semifields and Trombetti–Zhou codes to the $\ell_F>1$ regime. The authors prove MRD properties, compute nuclei/idealiser parameters, and establish the newness of the second family over finite fields, including semifield results, thereby broadening the landscape of MRD codes and nonassociative division algebras.
Abstract
We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods lead to the construction of new (not necessarily associative) division algebras and maximum rank distance (MRD) codes over both finite and infinite division rings. In particular, we construct new non-associative division algebras whose right nucleus is a central simple algebra having degree greater than 1. Over finite fields, we obtain new semifields and MRD codes for infinitely many choices of parameters. These families extend and contain many of the best previously known constructions.