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.

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....

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...

Lattice-inadmissible incidence structures

Frantisek Machala, Vladimír Slezák (2004)

Discussiones Mathematicae - General Algebra and Applications

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Join-independent and meet-independent sets in complete lattices were defined in [6]. According to [6], to each complete lattice (L,≤) and a cardinal number p one can assign (in a unique way) an incidence structure $J_{L}^{p}$ of independent sets of (L,≤). In this paper some lattice-inadmissible incidence structures are founded, i.e. such incidence structures that are not isomorphic to any incidence structure $J_{L}^{p}$ .

On covariety lattices

Tomasz Brengos (2008)

Discussiones Mathematicae - General Algebra and Applications

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This paper shows basic properties of covariety lattices. Such lattices are shown to be infinitely distributive. The covariety lattice $L_{C V} (K)$ of subcovarieties of a covariety K of F-coalgebras, where F:Set → Set preserves arbitrary intersections is isomorphic to the lattice of subcoalgebras of a $P_{κ}$ -coalgebra for some cardinal κ. A full description of the covariety lattice of Id-coalgebras is given. For any topology τ there exist a bounded functor F:Set → Set and a covariety K of F-coalgebras,...