  We study the classical problem of noisy constrained capacity in the case of
the binary symmetric channel (BSC), namely, the capacity of a BSC whose inputs
are sequences chosen from a constrained set. Motivated by a result of
Ordentlich and Weissman [In Proceedings of IEEE Information Theory Workshop
(2004) 117--122], we derive an asymptotic formula (when the noise parameter is
small) for the entropy rate of a hidden Markov chain, observed when a Markov
chain passes through a BSC. Using this result, we establish an asymptotic
formula for the capacity of a BSC with input process supported on an
irreducible finite type constraint, as the noise parameter tends to zero.
