On the construction of dense lattices with a given automorphisms group

Philippe Gaborit; Gilles Zémor

Displaying similar documents to “On the construction of dense lattices with a given automorphisms group”

Universal codes and unimodular lattices

Robin Chapman, Patrick Solé (1996)

Journal de théorie des nombres de Bordeaux

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Binary quadratic residue codes of length $p + 1$ produce via construction $B$ and density doubling type II lattices like the Leech. Recently, quaternary quadratic residue codes have been shown to produce the same lattices by construction $A$ modulo $4$ . We prove in a direct way the equivalence of these two constructions for $p \leq 31$ . In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice $B W_{32}$ . The equivalence between construction $B$ modulo $4$ plus density doubling...

Another 80-dimensional extremal lattice

Mark Watkins (2012)

Journal de Théorie des Nombres de Bordeaux

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We show that the unimodular lattice associated to the rank 20 quaternionic matrix group ${SL}_{2} (F_{41}) \otimes {\tilde{S}}_{3} \subset {GL}_{80} (Z)$ is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the $Θ$ -series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite simple groups) provides one proof that this lattice is distinct from the previous three, while computing the inner...

Characteristic vectors of unimodular lattices which represent two

Mark Gaulter (2007)

Journal de Théorie des Nombres de Bordeaux

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We improve the known upper bound of the dimension $n$ of an indecomposable unimodular lattice whose shadow has the third largest possible length, $n - 16$ .

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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The densest packing of 13 congruent circles in a circle.

Fodor, Ferenc (2003)

Beiträge zur Algebra und Geometrie

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On lower bounds of the density of Delone sets and holes in sequences of sphere packings.

Muraz, G., Verger-Gaugry, J.-L. (2005)

Experimental Mathematics

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Codes, lattices, and Steiner systems.

Solé, Patrick (1997)

The Electronic Journal of Combinatorics [electronic only]

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Identifying codes with small radius in some infinite regular graphs.

Charon, Irène, Hudry, Olivier, Lobstein, Antoine (2002)

The Electronic Journal of Combinatorics [electronic only]

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$2$ -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...