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

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.

Section 6 is expanded and a new Section 7 is added. Some small mistakes are corrected all over

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Cited by in corpus (5)

• How to obtain lattices from (f,sigma,delta)-codes via a generalization of Construction A
• How a nonassociative algebra reflects the properties of a skew polynomial
• Quotients of orders in algebras obtained from skew polynomials with applications to coding theory
• The automorphisms of generalized cyclic Azumaya algebras
• Solvable crossed product algebras revisited