Lower bounds on the anticipation of encoders for input-constrained channels

An input constrained channel S is de ned as the set of words generated by a nite labeled directed graph It is shown that every nite state encoder with nite antici pation i e with nite decoding delay for S can be obtained through state splitting rounds applied to some deterministic graph presentation of S followed by a reduction of equivalent states Furthermore each splitting round can be restricted to follow a certain prescribed structure This result in turn provides a necessary and su cient condition on the existence of nite state encoders for S with a given rate p q and a given anticipation a A second condition is derived on the existence of such encoders this condition is only necessary but it applies to every deterministic graph presentation of S Based on these two conditions lower bounds are derived on the anticipation of nite state encoders Those lower bounds improve on previously known bounds and in particular they are shown to be tight for the common rates used for the RLL and RLL constraints