A Note on Lattice Packings via Lattice Refinements

Rogers proved in a constructive way that every packing lattice Lambda of a symmetric convex body K in R-n is contained in a packing lattice whose covering radius is less than 3. By a slight modification of Rogers' approach better bounds for l(p) - balls are obtained. Together with Rogers' constructive proof, this leads, for instance, to a simple o(n(n/2)) running time algorithm that refines successively the packing lattice D-n (checkboard lattice) of the unit ball B-n and terminateswith a packing lattice (Lambda) over bar with density d((Lambda) over bar, B-n) > 2-1.197 n. We have also implemented this algorithm and in small dimensions (<= 25) and for certain simple structured start lattices like Z(n) or D-n the algorithm often terminates with packing lattices achieving the best-known lattice densities.