Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes

Pumpluen, Susanne

Abstract

Let S be a unital ring, S[t; σ, δ] a skew polynomial ring where σ is an injective endomorphism and δ a left σ -derivation, and suppose f ε S[t; σ, δ] has degree m and an invertible leading coefficient. Using right division by f to define the multiplication, we obtain unital nonassociative algebras Sf on the set of skew polynomials in S[t; σ, δ] 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 Sf can be employed both to design linear (f, σ, δ)-codes over unital rings and to study their behaviour.

Citation

Pumpluen, S. (2017). Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes. Advances in Mathematics of Communications, 11(3), 615-634. https://doi.org/10.3934/amc.2017046

• Journal Article Type: Article
• Acceptance Date: Jul 1, 2016
• Publication Date: Aug 1, 2017
• Deposit Date: Nov 18, 2016
• Publicly Available Date: Aug 2, 2018
• Journal: Advances in Mathematics of Communications
• Print ISSN: 1930-5346
• Electronic ISSN: 1930-5338
• Publisher: American Institute of Mathematical Sciences (AIMS)
• Peer Reviewed: Peer Reviewed
• Volume: 11
• Issue: 3
• Pages: 615-634
• DOI: https://doi.org/10.3934/amc.2017046
• Keywords: Skew Polynomial Ring, Ore Polynomials, Nonassociative Algebra, Commutative Finite Chain Ring,; Generalized Galois Rings, Linear Codes, (f, σ, δ)-codes, Skew-constacyclic Codes
• Public URL: https://nottingham-repository.worktribe.com/output/875867
• Publisher URL: http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=14505
• Additional Information: This is a pre-copy-editing, author-produced PDF of an article accepted for publication in [insert journal title] following peer review. The definitive publisher-authenticated version is available online at: http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=14505.
• Contract Date: Nov 18, 2016