In this letter, we discussed some properties of characteristic generators for a finite Abelian group code, proved that any two characteristic generators can not start (end) at the same position and have the same order of the starting (ending) components simultaneously, and that the number of all characteristic generators can be directly computed from the group code itself. These properties are exactly the generalization of the corresponding trellis properties of a linear code over a field.

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.

Tail-biting trellises of linear and nonlinear block codes are addressed. We refine the information-theoretic approach of a previous work on conventional trellis representation, and show that the same ideas carry over to tail-biting trellises. We present lower bounds on the state and branch complexity profiles of these representations. These bounds are expressed in terms of mutual information between different portions of the code, and they introduce the notions of superstates and superbranches. For linear block codes, our bounds imply that the total number of superstates, and respectively superbranches, of a tail-biting trellis of the code cannot be smaller than the total number of states, and respectively branches, of the corresponding minimal conventional trellis, though the total number of states and branches of a tail-biting trellis is usually smaller than that of the conventional trellis. We also develop some improved lower bounds on the state complexity of a tail-biting trellis for two classes of codes: the first-order Reed-Muller codes and cyclic codes. We show that the superstates and superbranches determine the Viterbi decoding complexity of a tail-biting trellis. Thus, the computational complexity of the maximum-likelihood decoding of linear block codes on a tail-biting trellis, using the Viterbi algorithm, is not smaller than that of the conventional trellis of the code. However, tail-biting trellises are beneficial for suboptimal and iterative decoding techniques.