Universal codes and unimodular lattices

Robin Chapman; Patrick Solé

Displaying similar documents to “Universal codes and unimodular lattices”

$2$ -modular lattices from ternary codes

Robin Chapman, Steven T. Dougherty, Philippe Gaborit, Patrick Solé (2002)

Journal de théorie des nombres de Bordeaux

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The alphabet $𝐅_{3} + v 𝐅_{3}$ where $v^{2} = 1$ is viewed here as a quotient of the ring of integers of $𝐐 (\sqrt{- 2})$ by the ideal (3). Self-dual $𝐅_{3} + v 𝐅_{3}$ codes for the hermitian scalar product give $2$ -modular lattices by construction $A_{K}$ . There is a Gray map which maps self-dual codes for the Euclidean scalar product into Type III codes with a fixed point free involution in their automorphism group. Gleason type theorems for the symmetrized weight enumerators of Euclidean self-dual codes and the length weight enumerator of hermitian...

On extremal additive $𝔽_{4}$ codes of length $10$ to $18$

Christine Bachoc, Philippe Gaborit (2000)

Journal de théorie des nombres de Bordeaux

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In this paper we consider the extremal even self-dual $𝔽_{4}$ -additive codes. We give a complete classification for length $10$ . Under the hypothesis that at least two minimal words have the same support, we classify the codes of length $14$ and we show that in length $18$ such a code is equivalent to the unique $𝔽_{4}$ -hermitian code with parameters [18,9,8]. We construct with the help of them some extremal $3$ -modular lattices.

Chinese TeX Typesetting: Past and Present

Jiang Jiang (2010)

Zpravodaj Československého sdružení uživatelů TeXu

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The article introduces and overviews Chinese TeX typesetting from its early beginnings to the present day.

On kissing numbers in dimensions 32 to 128.

Edel, Yves, Rains, E.M., Sloane, N.J.A. (1998)

The Electronic Journal of Combinatorics [electronic only]

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Codes, lattices, and Steiner systems.

Solé, Patrick (1997)

The Electronic Journal of Combinatorics [electronic only]

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Exploring invariant linear codes through generators and centralizers

Partha Pratim Dey (2005)

Archivum Mathematicum

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We investigate a $H$ -invariant linear code $C$ over the finite field $F_{p}$ where $H$ is a group of linear transformations. We show that if $H$ is a noncyclic abelian group and $(| H |, p) = 1$ , then the code $C$ is the sum of the centralizer codes $C_{c} (h)$ where $h$ is a nonidentity element of $H$ . Moreover if $A$ is subgroup of $H$ such that $A ≅ Z_{q} \times Z_{q}$ , $q \neq p$ , then dim $C$ is known when the dimension of $C_{c} (K)$ is known for each subgroup $K \neq 1$ of $A$ . In the last few sections we restrict our scope of investigation to a special class of invariant codes,...

On the construction of dense lattices with a given automorphisms group

Philippe Gaborit, Gilles Zémor (2007)

Annales de l’institut Fourier

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We consider the problem of constructing dense lattices in $ℝ^{n}$ with a given non trivial automorphisms group. We exhibit a family of such lattices of density at least $c n 2^{- n}$ , which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions $n$ , we exhibit a finite set of lattices that come with an automorphisms group of size $n$ , and a constant proportion of which achieves the aforementioned lower bound on the largest packing density....

An extension of the ordering based on nullnorms

Emel Aşıcı (2019)

Kybernetika

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In this paper, we generally study an order induced by nullnorms on bounded lattices. We investigate monotonicity property of nullnorms on bounded lattices with respect to the $F$ -partial order. Also, we introduce the set of incomparable elements with respect to the F-partial order for any nullnorm on a bounded lattice. Finally, we investigate the relationship between the order induced by a nullnorm and the distributivity property for nullnorms.

Construction methods for uni-nullnorms and null-uninorms on bounded lattice

Ümit Ertuğrul, M. Nesibe Kesicioğlu, Funda Karaçal (2019)

Kybernetika

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In this paper, two construction methods have been proposed for uni-nullnorms on any bounded lattices. The difference between these two construction methods and the difference from the existing construction methods have been demonstrated and supported by an example. Moreover, the relationship between our construction methods and the existing construction methods for uninorms and nullnorms on bounded lattices are investigated. The charactertics of null-uninorms on bounded lattice $L$ are...

On a generalization of Craig lattices

Hao Chen (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range $3332 - 4096$ which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range $128 - 3272$ . We also construct some dense lattices of dimensions in the range $4098 - 8232$ . Finally we also obtain some new lattices of moderate dimensions such as $68, 84, 85, 86$ , which are denser...