Mathematics – Number Theory
Scientific paper
2006-05-03
Annales de l'institut Fourier, 57 no. 4 (2007), p. 1051-1062
Mathematics
Number Theory
10 pages. Corrected typos and ambiguous definition
Scientific paper
We consider the problem of constructing dense lattices of R^n with a given automorphism group. We exhibit a family of such lattices of density at least cn/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 automorphism group of size n, and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic complexity for exhibiting a basis of such a lattice is of order exp(nlogn), which improves upon previous theorems that yield an equivalent lattice packing density. The method developed here involves applying Leech and Sloane's construction A to a special class of codes with a given automorphism group, namely the class of double circulant codes.
Gaborit Philippe
Zemor Gilles
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