Upper bounds on trellis complexity of lattices

Abstract

Unlike block codes, n-dimensional lattices can have minimal trellis diagrams with an arbitrarily large number of states, branches, and paths. In particular, we show by a counterexample that there is no f(n), a function of n, such that all rational lattices of dimension n have a trellis with less than f(n) states. Nevertheless, using a theorem due to Hermite, we prove that every integral lattice /spl Lambda/ of dimension n has a trellis T, such that the total number of paths in T is upper-bounded by P(T)/spl les/n!(2//spl radic/3)/sup n2(n-1/2)/V(/spl Lambda/)/sup n-1/ where V(n) is the volume of /spl Lambda/. Furthermore, the number of states at time i in T is upper-bounded by |S/sub i/|/spl les/(2//spl radic/3)/sup i2(n-1)/V(/spl Lambda/)/sup 2i2/n/. Although these bounds are seldom tight, these are the first known general upper bounds on trellis complexity of lattices.