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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

The alphabet 𝐅 3 + v 𝐅 3 where v 2 = 1 is viewed here as a quotient of the ring of integers of 𝐐 ( - 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 self-dual...

On a generalization of Craig lattices

Hao Chen (2013)

Journal de Théorie des Nombres de Bordeaux

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 than the...

On the construction of dense lattices with a given automorphisms group

Philippe Gaborit, Gilles Zémor (2007)

Annales de l’institut Fourier

We consider the problem of constructing dense lattices in 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. The algorithmic...

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