On the construction of dense lattices with a given automorphisms group
Philippe Gaborit, Gilles Zémor · Annales de l’institut Fourier · 2007
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. The algorithmic complexity for exhibiting a basis of such a lattice is of order exp ( n log n ) , 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.