For every positive integer n, the classical sphere packing problem asks for the densest arrangement of spheres in Rn. Despite important advances, the general problem remains open. The present work concerns the particular case of lattice packings. Craig's family of lattice packings includes the majority of the densest packings known in dimensions 148 to 3000. In addition, the lattices are asymptotically good with respect to density where "good" means that densities satisfy a certain lower bound. Craig's lattices however, only exist in dimensions p -- 1 where p is a prime. The purpose of this work is to bring to light a generalization and refinement of Craig's family. Not only do we present a lattice in every dimension n for any positive integer n, but when n = p -- 1, with p a prime, the associated lattice has a density which is several times greater than the density of the original Craig's lattice.