Trellis diagrams of lattices and the Viterbi algorithm can be used for decoding. It has been known that the numbers of states and labels at every level of any finite trellis diagrams of a lattice L and its dual L* under the same coordinate system are the same. In the paper, we present concrete expressions of the numbers of distinct paths in the trellis diagrams of L and L* under the same coordinate system, which are more concrete than Theorem 2 of [1]. We also give a relation between the numbers of edges in the trellis diagrams of L and L*. Furthermore, we provide the upper bounds on the state numbers of a trellis diagram of the lattice L1L2 by the state numbers of trellis diagrams of lattices L1 and L2.

It is well known that the trellises of lattices can be employed to decode efficiently. It was proved in [1] and [2] that if a lattice L has a finite trellis under the coordinate system , then there must exist a basis (b1,b2,,bn) of L such that Wi=span() for 1in. In this letter, we prove this important result in a completely different method, and give an efficient method to compute all bases of this type.