Lower bounds on trellis complexity of block codes

The trellis state-complexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis. We present a new lower bound on the state complexity of linear codes, which includes most of the existing bounds as special cases. The new bound is obtained by dividing the time axis for the code into several sections of varying lengths, as opposed to the division into two sections-the past and the future-employed in the well-known DLP bounds. For a large number of codes this results in a considerable improvement upon the DLP bound. Moreover, we generalize the new bound to nonlinear codes, and introduce several alternative techniques for lower-bounding the trellis complexity, based on the distance spectrum and other combinatorial properties of the code. We also show how the general ideas developed in this paper may be employed to lower-bound the maximum and the total number of branches in the trellis, leading to considerably tighter bounds on these quantities. Furthermore, the asymptotic behavior of the new bounds is investigated, and shown to improve upon the previously known asymptotic estimates of trellis state-complexity.