Density/length profiles and trellis complexity of lattices

The density/length profile (DCP) of a lattice /spl Lambda/ is analogous to the dimension/length profile of a linear code. The DLP is a geometrical invariant of /spl Lambda/ that includes the coding gain of /spl Lambda/. Duality results analogous to those of linear block codes are derived for lattices. Bounds on the DLP may be derived from bounds on Hermite's constants; these hold with equality for many dense lattices. In turn, the DLP lowerbounds the state complexity profile of a minimal trellis diagram for /spl Lambda/ in any coordinate system. It is shown that this bound can be met for the E/sub 8/ lattice by a laminated lattice construction with a novel trellis diagram. Bounds and constructions for other important low-dimensional lattices are given. Two laminated lattice constructions of the Leech lattice yield trellis diagrams with maximum state space sizes 1024 and 972.<
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