On the construction of dense lattices with a given automorphism group

We consider the problem of constructing dense lattices of Rn with a given non trivial automorphism 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 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(n logn), 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.