Maximum minimal distance partitioning of the /spl Zopf//sup 2/ lattice

We study the problem of dividing the /spl Zopf//sup 2/ lattice into partitions so that minimal intra-partition distance between the points is maximized. We show that this problem is analogous to the problem of sphere packing. An upper bound on the achievable intra-partition distances for a given number of partitions follows naturally from this observation, since the optimal sphere packing in two dimensions is achieved by the hexagonal lattice. Specific instances of this problem, when the number of partitions is 2/sup m/, were treated in trellis-coded modulation (TCM) code design by Ungerboeck (1982) and others. It is seen that methods previously used for set partitioning in TCM code design are asymptotically suboptimal as the number of partitions increases. We propose an algorithm for solving the /spl Zopf//sup 2/ lattice partitioning problem for an arbitrary number of partitions.