Bounds on the number of states in encoders graphs for input-constrained channels, IEEE Trans. Canonical encoders for sliding block decoders, submitted for publication. Variable-length state splitting with applications to average runlength-constrained (ARC) codes, IEEE Trans. By 6]]7], the existence of such a left-resolving conjugacy from paths in E to paths in G implies that E is obtained from G by a sequence of state splittings consistent with respect to the vector x deened by x u = number states in E which map to u: Conversely, if E is obtained by a sequence of state splittings, starting from G, then E is lossless of nite order. It is not hard to modify this example to produce a constrained system S = S(G) with c(S) > log 3 and a certain (minimal) (A G ; 3)-approximate eigenvector x such that no steth-ering (S; 3)-encoder based on x can possibly be lossless of nite order. But we do not know of an example where c(S) > log 3 and where we can prove that there is no stethering (S; 3)-encoder which is lossless of nite order. Acknowledgment The authors thank Moni Naor for helpful discussions. smallest (A G ; 3)-eigenvector. Now, by exhaustive search, one can check all of the stethering (S; 3)-encoders based on x (there are 72 of them) and see that none of these is lossless of nite order (an alternative argument, which requires less search, is sketched below). Therefore , by Proposition 10, there can be no stethering (S; 3)-encoder which is lossless of nite order. We now sketch the alternative argument. The state-splitting algorithm (see 1]]22]) constructs encoders which are lossless of nite order (in fact, for nite memory constraints, it constructs encoders which are sliding-block decodable). This algorithm proceeds by splitting states of a deterministic presentation G in an iterative fashion until one arrives at a new presentation that is suitable for use as an encoder. The splittings are guided by choices based on an (A G ; n)-approximate eigenvector x, and we then say that the splittings are consistent with respect to x. As in stethering, the states of the encoders are descendants (u; i); 0 i < x u of states u in G. It is straightforward to see that, for Example 7, the rst three rounds of any sequence of state splittings, consistent with respect to the smallest eigenvector x = 2 4 …