The extended golay codes considered as ideals

Bernhardt, Frank; Landrock, Peter; Manz, Olaf

doi:10.1016/0097-3165(90)90069-9

Cited by 41 publications

(27 citation statements)

References 6 publications

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“…Initially, it was shown in [1] that the [24, 12,8] could be constructed from ideals in the group algebra F 2 S 4 where S 4 is the symmetric group on 4 elements. In [14], the [24, 12,8] code was constructed from F 2 D 24 .…”

Section: Binary Golay Codementioning

confidence: 99%

Group rings, G-codes and constructions of self-dual and formally self-dual codes

Dougherty

Gildea

Taylor

et al. 2017

Des. Codes Cryptogr.

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We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in [11] by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes correspond to ideals in the group ring RG and as such must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36,16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48.

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Section: Binary Golay Codementioning

confidence: 99%

Group rings, G-codes and constructions of self-dual and formally self-dual codes

Dougherty

Gildea

Taylor

et al. 2017

Des. Codes Cryptogr.

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“…This is a [21,6,8]-code, which has the same weight of the best known [21,6]-code (see [16]). If q = 11, a direct computation show us that w(I) = 5, the weight of the best known [6,2]-code according to [16]. This is also the case for any field of characteristic different from 2, 3, 5 and 7.…”

Section: Non Abelian Groupsmentioning

confidence: 99%

“…In this very short survey we cover only one aspect of this role: we focus on the relationship between weight and dimension of group codes. This type of codes have recently been the object of active research (see [2], [6], [7], [8], [9], [10], [11], [12], [13], [19], [20], [24], [27]).…”

Section: Introductionmentioning

confidence: 99%

Group algebras and coding theory: a short survey

Milies¹

2019

revint

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We study codes constructed from ideals in group algebras and we are particularly interested in their dimensions and weights. First we introduced a special kind of idempotents and study the ideals they generate. We use this information to show that there exist abelian non-cyclic groups that give codes which are more convenient than the cyclic ones. Finally, we discuss briefly some facts about non-abelian codes. Álgebras de grupo y teoría de códigos: una breve reseñaResumen. Estudiamos códigos construidos a partir de ideales de álgebras de grupo y estamos particularmente interesados en sus dimensiones y pesos. Introducimos inicialmente un tipo especial de idempotentes y estudiamos los ideales que generan. Usamos esta información para mostrar que existen grupos abelianos no cíclicos que son más convenientes que los cíclicos. Finalmente, discutimos brevemente algunos resultados sobre códigos no abelianos.

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“…any F X-submodule of the regular F X-module, is called a group code of the group X over the field F. The study on group codes has been there since many years, e.g. [2]. In recent years it has attracted attentions to explore the conditions for the existence of self-dual group codes.…”

Section: Introductionmentioning

confidence: 99%

On the existence of self-dual permutation codes of finite groups

Fan

Zhang

2011

Des. Codes Cryptogr.

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Motivated by a research on self-dual extended group codes, we consider permutation codes obtained from submodules of a permutation module of a finite group of odd order over a finite field, and demonstrate that the condition "the extension degree of the finite field extended by n'th roots of unity is odd" is sufficient but not necessary for the existence of self-dual extended transitive permutation codes of length n + 1. It exhibits that the permutation code is a proper generalization of the group code, and has more delicate structure than the group code.

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The extended golay codes considered as ideals

Cited by 41 publications

References 6 publications

Group rings, G-codes and constructions of self-dual and formally self-dual codes

Group rings, G-codes and constructions of self-dual and formally self-dual codes

Group algebras and coding theory: a short survey

On the existence of self-dual permutation codes of finite groups

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