  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.
