Quotients of orders in algebras obtained from skew polynomials with applications to coding theory

TL;DR

This paper explores nonassociative finite rings derived from skew polynomial-based algebras, extending their application in coding theory by analyzing quotients of natural orders and their use in coset coding.

Contribution

It introduces new families of nonassociative rings from skew polynomial algebras and demonstrates their application in coset coding, generalizing previous associative algebra results.

Findings

Families of nonassociative rings from skew polynomial algebras are characterized.

Quotients of natural orders can be effectively used for coset coding.

Previous associative algebra results are recovered as special cases.

Abstract

We describe families of nonassociative finite unital rings that occur as quotients of natural nonassociative orders in generalized nonassociative cyclic division algebras over number fields. These natural orders have already been used to systematically construct fully diverse fast-decodable space-time block codes. We show how the quotients of natural orders can be employed for coset coding. Previous results by Oggier and Sethuraman involving quotients of orders in associative cyclic division algebras are obtained as special cases.