Quotients of skew polynomial rings: new constructions of division algebras and MRD codes

TL;DR

This paper introduces novel constructions of division algebras and MRD codes using quotients of skew polynomial rings, including explicit examples and extensions of known families over finite and infinite division rings.

Contribution

It provides the first explicit example where the ratio of degrees in skew polynomial rings is non-extremal, leading to new non-associative division algebras and MRD codes.

Findings

Constructed new non-associative division algebras with larger right nuclei.

Developed new MRD codes over finite and infinite division rings.

Extended known families of semifields and MRD codes with many parameter choices.

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.