Lattice Encoding of Cyclic Codes from Skew-Polynomial Rings

Ducoat, Jérôme; Oggier, Frédérique

doi:10.1007/978-3-319-17296-5_16

Cited by 6 publications

(6 citation statements)

References 13 publications

(9 reference statements)

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“…, n − 1}. We prove that these are the only automorphisms of S f : a 0 = 0 and so N K/F (k) = 1 by (9). Therefore, by Hilbert 90, there exists α ∈ K such that k = σ(α)/α.…”

Section: Cyclic Subgroups Of Autmentioning

confidence: 87%

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The automorphisms of Petit’s algebras

Brown

Pumpluen

2017

Communications in Algebra

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Abstract. Let σ be an automorphism of a field K with fixed field F . We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t; σ]/f K[t; σ] obtained when the twisted polynomial f ∈ K[t; σ] is invariant, and were first defined by Petit. We compute all their automorphisms if σ commutes with all automorphisms in Aut F (K) and n ≥ m − 1, where n is the order of σ and m the degree of f , and obtain partial results for n < m − 1. In the case where K/F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F . We also briefly investigate when two such algebras are isomorphic.

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“…, n − 1}. We prove that these are the only automorphisms of S f : a 0 = 0 and so N K/F (k) = 1 by (9). Therefore, by Hilbert 90, there exists α ∈ K such that k = σ(α)/α.…”

Section: Cyclic Subgroups Of Autmentioning

confidence: 87%

“…Moreover, recently space-time block codes, coset codes and wire-tap codes were obtained employing the algebras S f over number fields, cf. [9,10,21,24,27,28,29], and they also appear useful for linear cyclic codes [25,26].…”

Section: Introductionmentioning

confidence: 99%

The automorphisms of Petit’s algebras

Brown

Pumpluen

2017

Communications in Algebra

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“…Our approach canonically generalizes and unifies the ones of [5] and [6]: the situation considered there only deals with natural orders in associative cyclic division algebras, i.e. where f = t m − d ∈ O K [t; σ] is irreducible and K/F is a cyclic number field extension of degree m with Galois group generated by σ.…”

Section: Discussionmentioning

confidence: 99%

“…The algebras A = (D, σ, d) are behind the fast-decodable iterated codes in [14], [16], [8] [19]. Equation (5) and the isomorphism in (6) imply that the right multiplication in Λ/IΛ is given by the mn × mn matrix in (7) where the entries are read modulo IO K . We call this matrix M (x).…”

Section: Quotients Of Natural Orders In S F Imentioning

confidence: 99%

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

Pumplün

2018

Communications in Algebra

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We describe families of nonassociative finite unital rings that occur as quotients of natural nonassociative orders in generalized nonassociative cyclic division algebras over number fields. These natural orders have already been used to systematically construct fully diverse fast-decodable space-time block codes. We show how the quotients of natural orders can be employed for coset coding. Previous results by Oggier and Sethuraman involving quotients of orders in associative cyclic division algebras are obtained as special cases.2010 Mathematics Subject Classification. Primary: 17A35; Secondary: 11T71, 94B40, 94B05.

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“…The algebras S f were previously introduced by Petit, but only for the case that S is a division ring, hence S[t; σ, δ] left and right Euclidean [33]. In that setting, they already appeared in [12], [13], [32], and were used in space-time block coding, cf. [47], [38], [39].…”

Section: Introductionmentioning

confidence: 99%

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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Lattice Encoding of Cyclic Codes from Skew-Polynomial Rings

Cited by 6 publications

References 13 publications

The automorphisms of Petit’s algebras

The automorphisms of Petit’s algebras

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

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

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