Perfect codes in Euclidean lattices

In the present paper, we investigate the existence of lattice perfect codes when considered as sublattices of other lattices under the Euclidean metric. We discuss the connection between a discrete tiling of a lattice and a continuous tilling of the n-dimensional space and equivalent characterizations of discrete tilings. We generalize bounds on the radius of perfect codes in a generic lattice, which were previously known for the cubic lattice, and we provide some new bounds. The new bounds are based on the packing and the covering densities and on the covering radius of the ambient lattice. An algorithm is presented for the search of perfect codes which is used to derive all perfect codes for a collection of ambient lattices in dimensions two and three. In contrast to the cubic lattice, these case studies show that by considering general ambient lattices one can find rich sets of perfect codes.