Trellis complexity versus the coding gain of lattices. II

For pt.I see ibid., vol. 42, no.6, p.1796-1802, 1996. Every rational lattice has a finite trellis diagram which can be employed for maximum-likelihood decoding over the additive white Gaussian noise channel via the Viterbi algorithm. For an arbitrary rational lattice L with gain γ, the average number of states (respectively, branches) in any given trellis diagram of L is bounded below by a function of γ. It is proved that this function grows exponentially in γ. In the reverse direction, it is proved that given ∈>0, for arbitrarily large values of γ, there exist lattices of gain γ with an average number of branches and states less than exp(γ^(1+∈)). Trellis diagrams of block codes obtained from truncated convolutional codes are employed to show that, inside the trellis model, the problem of decoding lattices is not much harder than exponential