Minimal trellis diagrams of lattices

Abstract

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.