Minimal trellis diagrams of lattices

Unlike block codes, the number of distinct paths on the trellis diagrams of lattices (N) depends highly on the selected (trellis) coordinate system. Focusing on N as the measure of complexity, it is shown that the problem of finding a proper trellis of a lattice can be reduced to the problem of finding a proper basis for the lattice. For a lattice /spl Lambda/ with coding gain /spl gamma/ and dimension n, a lower bound of the form [/spl gamma//sup n/2/] on N is derived. A trellis of /spl Lambda/ is called minimal if it achieves the lower bound (or more generally, if it minimizes N). For many important lattices like Barnes-Wall lattices BW/sub n/, n=2/sup m/, the Leech lattice /spl Lambda//sub 24/, D/sub n/, D/sub n//sup /spl perp//, E/sub n/, E/sub n//sup /spl perp//, and A/sub n/, n/spl les/9, we obtain the basis matrices which result in minimal trellis diagrams. For some other lattices like the Coxeter-Todd lattice K/sub 12/, and A/sub n/, A/sub n//sup /spl perp//, n>9, trellises with small values of N (probably not minimal) are obtained. The constructed trellises, which are novel in many cases, can be employed to efficiently decode the lattices via the Viterbi algorithm.