Finite nonassociative algebras obtained from skew polynomials and possible applications to $(f,\sigma,\delta)$-codes

TL;DR

This paper constructs and analyzes finite nonassociative algebras derived from skew polynomial rings, exploring their structure and applications to coding theory over rings.

Contribution

It introduces a new class of nonassociative algebras from skew polynomials and demonstrates their use in designing and studying $(f,\sigma,\delta)$-codes over rings.

Findings

Finite nonassociative algebras $S_f$ are constructed from skew polynomials.

When $S$ is a Galois ring and $f$ is irreducible, $S_f$ forms finite rings with specific zero divisor structures.

These algebras can be used to design and analyze linear $(f,\sigma,\delta)$-codes over rings.

Abstract

Let $S$ be a unital ring, $S[t;\sigma,\delta]$ a skew polynomial ring where $\sigma$ is an injective endomorphism and $\delta$ a left $\sigma$-derivation, and suppose $f\in S[t;\sigma,\delta]$ has degree $m$ and an invertible leading coefficient. Using right division by $f$ to define the multiplication, we obtain unital nonassociative algebras $S_f$ on the set of skew polynomials in $S[t;\sigma,\delta]$ of degree less than $m$. We study the structure of these algebras. When $S$ is a Galois ring and $f$ base irreducible, these algebras yield families of finite unital nonassociative rings $A$, whose set of (left or right) zero divisors has the form $pA$ for some prime $p$. For reducible $f$, the $S_f$ can be employed both to design linear $(f,\sigma,\delta)$-codes over unital rings and to study their behaviour.