Trellis complexity versus the coding gain of lattices. II

Abstract

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 /spl gamma/, the average number of states (respectively, branches) in any given trellis diagram of L is bounded below by a function of /spl gamma/. It is proved that this function grows exponentially in /spl gamma/. In the reverse direction, it is proved that given /spl isin/>0, for arbitrarily large values of /spl gamma/, there exist lattices of gain /spl gamma/ with an average number of branches and states less than exp(/spl gamma//sup (1+/spl isin//)). 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.