Bounds on the number of states in encoder graphs for input-constrained channels

We exhibit lower bounds on the number of states in a fixed rate finite-state encoder that maps unconstrained n-ary sequences into a given set of constrained sequences, defined by a finite labelled graph G. In particular, one simple lower bound is given by minxmaxv xv where x = [xv] ranges over certain (nonnegative integer) approximate eigenvectors of the adjacency matrix for G. In some sense, our bounds are close to what can be realized by the state splitting algorithm, and in some cases, they are shown to be tight. In particular, these bounds are used to show that the smallest (in number of states) known encoders for the (1, 7) and (2, 7) run-length-limited systems are indeed the smallest possible. For any given constrained set S of sequences, we apply these bounds to study the growth of the number of states in families of encoders whose rates approach the capacity of S.