Density/length profiles and trellis complexity of lattices

The density/length profile (DCP) of a lattice Λ is analogous to the dimension/length profile of a linear code. The DLP is a geometrical invariant of Λ that includes the coding gain of Λ. 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 Λ in any coordinate system. It is shown that this bound can be met for the E₈ 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