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

Pumpluen, Susanne

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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.

Journal Article Type Article
Publication Date Aug 1, 2017
Journal Advances in Mathematics of Communications
Print ISSN 1930-5346
Electronic ISSN 1930-5338
Publisher American Institute of Mathematical Sciences
Peer Reviewed Peer Reviewed
Volume 11
Issue 3
APA6 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), https://doi.org/10.3934/amc.2017046
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
Publisher URL http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=14505
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf
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....snew.jsp?paperID=14505.

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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