On the algebraic structure of quasi group codes

Borello, Martino; Willems, Wolfgang M.

doi:10.48550/arxiv.1912.09167

Cited by 3 publications

(3 citation statements)

References 12 publications

(19 reference statements)

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“…As observed in [8], a linear code of length 2m can be seen as a D 2m -code if and only if its automorphism group contains a subgroup isomorphic to D 2m all of whose nontrivial elements act fixed point free on the coordinates {1, . .…”

Section: Dihedral Codesmentioning

confidence: 99%

Dihedral codes with prescribed minimum distance

Borello¹,

Jamous²

2020

Preprint

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Dihedral codes, particular cases of quasi-cyclic codes, have a nice algebraic structure which allows to store them efficiently. In this paper, we investigate it and prove some lower bounds on their dimension and minimum distance, in analogy with the theory of BCH codes. This allows us to construct dihedral codes with prescribed minimum distance. In the binary case, we present some examples of optimal dihedral codes obtained by this construction.

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Section: Dihedral Codesmentioning

confidence: 99%

Dihedral codes with prescribed minimum distance

Borello¹,

Jamous²

2020

Preprint

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“…Self-dual codes over these rings were studied in [17]. Recently, in [1], the authors study the algebraic structure of quasi-group codes.…”

Section: Quasi Composite G-codesmentioning

confidence: 99%

Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes

Dougherty¹,

Gildea²,

Korban³

et al. 2020

Preprint

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In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We define quasi-composite G-codes and give a construction of these codes. We also study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary selfdual codes of length 68 with new weight enumerators for the rare parameters γ = 7, 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.

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“…The algebraic structure of G-codes has been intensively studied; see e.g. [4,6,8,14] and the references therein. Yet, there are still many open questions about their coding theoretical properties.…”

Section: Introductionmentioning

confidence: 99%

On ideals in group algebras: an uncertainty principle and the Schur product

Borello¹,

Willems²,

Zini³

2022

Preprint

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In this paper we investigate some properties of ideals in group algebras of finite groups over fields. First, we highlight an important link between their dimension, their minimal Hamming distance and the group order. This is a generalized version of an uncertainty principle shown in 1992 by Meshulam. Secondly, we introduce the notion of the Schur product of ideals in group algebras and investigate the module structure and the dimension of the Schur square. We give a structural result on ideals that coincide with their Schur square, and we provide conditions for an ideal to be such that its Schur square has the projective cover of the trivial module as a direct summand. This has particularly interesting consequences for group algebras of p-groups over fields of characteristic p.

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On the algebraic structure of quasi group codes

Cited by 3 publications

References 12 publications

Dihedral codes with prescribed minimum distance

Dihedral codes with prescribed minimum distance

Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes

On ideals in group algebras: an uncertainty principle and the Schur product

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