On the construction of dense lattices with a given automorphisms group
Annales de l'Institut Fourier, Volume 57 (2007) no. 4, p. 1051-1062

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 cn2 -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 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 automorphisms group, namely the class of double circulant codes.

On s’intéresse à la construction de réseaux denses de n contenant un groupe d’automorphismes donné non trivial. On obtient une telle construction de réseaux, dont la densité est au moins cn2 -n , ce qui, à une constante multiplicative près, atteint la meilleure densité asymptotique connue d’un empilement de sphères. Plus précisément, on exhibe, pour une suite infinie de dimensions n, un ensemble de réseaux de groupe d’automorphismes fixé et de taille n, et dont une proportion constante atteint la borne inférieure précitée sur la densité. La complexité algorithmique de la construction d’une base d’un tel réseau dense est d’ordre exp(nlogn), ce qui améliore la complexité des constructions déjà connues de réseaux d’une densité équivalente. La méthode que nous proposons utilise la construction A de Leech et Sloane appliquée à une classe particulière de codes  : la classe des codes doublement circulants.

DOI : https://doi.org/10.5802/aif.2286
Classification:  06B20,  03G10
Keywords: Lattice packings, Minkowski-Hlawka lower bound, probability, automorphism group, double circulant codes
     author = {Gaborit, Philippe and Z\'emor, Gilles},
     title = {On the construction of dense lattices with a given automorphisms group},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {4},
     year = {2007},
     pages = {1051-1062},
     doi = {10.5802/aif.2286},
     mrnumber = {2339326},
     zbl = {1172.11021},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_4_1051_0}
Gaborit, Philippe; Zémor, Gilles. On the construction of dense lattices with a given automorphisms group. Annales de l'Institut Fourier, Volume 57 (2007) no. 4, pp. 1051-1062. doi : 10.5802/aif.2286. http://www.numdam.org/item/AIF_2007__57_4_1051_0/

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