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

Pumplün, S.

doi:10.1080/00927872.2018.1461882

Cited by 5 publications

(5 citation statements)

References 19 publications

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“…We also canonically obtain coset codes from orders in nonassociative algebras over number fields which are used for fast-decodable space-time block codes [34]. Again, previous results for coset codes related to associative cyclic algebras S f by Oggier and Sethuraman [32] are obtained as special cases.…”

Section: Conclusion and Further Workmentioning

confidence: 80%

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

Pumplün¹

2017

Advances in Mathematics of Communications

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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 ∈ 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 S f 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 S f can be employed both to design linear (f, σ, δ)-codes over unital rings and to study their behaviour.2010 Mathematics Subject Classification. Primary: 17A60; Secondary: 94B05.

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Section: Conclusion and Further Workmentioning

confidence: 80%

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

Pumplün¹

2017

Advances in Mathematics of Communications

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“…We construct these MRD codes using skew polynomials. Skew polynomials have been successfully used in constructions of both division algebras (mostly semifields) and linear codes [2,4,5,13,[26][27][28], in particular building space-time block codes (STBCs) [29] and MRD codes [33,34].…”

Section: Introductionmentioning

confidence: 99%

Division algebras and MRD codes from skew polynomials

Thompson¹,

Pumpluen

2023

Glasgow Math. J.

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Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\;\sigma,\delta ]$ a skew polynomial ring. Using skew polynomials $f\in R$ , we construct division algebras and maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include Jha Johnson semifields, and the classes of classical and twisted Gabidulin codes constructed by Sheekey.

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“…Skew polynomials have been successfully used in constructions of division algebras (mostly semifields) and linear codes [2,3,4,11,21,22,23], in particular building space-time block codes (STBCs) [24] and maximum rank distance (MRD) codes [27,28].…”

Section: Introductionmentioning

confidence: 99%

Division algebras and MRD codes from skew polynomials

Thompson¹,

Pumpluen²

2021

Preprint

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Let D be a division algebra, finite-dimensional over its center, and R = D[t; σ, δ] a skew polynomial ring.Using skew polynomials f ∈ R, we construct division algebras and a generalization of maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include a class of codes constructed by Sheekey (in particular, generalized Gabidulin codes), as well as Jha Johnson semifields.

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

Cited by 5 publications

References 19 publications

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

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

Division algebras and MRD codes from skew polynomials

Division algebras and MRD codes from skew polynomials

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