Asymptotics of input-constrained binary symmetric channel capacity

TL;DR

This paper derives an asymptotic formula for the capacity of a binary symmetric channel with input constraints, focusing on the low-noise regime, by analyzing the entropy rate of a hidden Markov process.

Contribution

It introduces a novel asymptotic formula for the constrained capacity of a BSC in the small-noise limit, extending previous results to input processes constrained by finite type conditions.

Findings

Derived an asymptotic capacity formula for low-noise BSC with input constraints

Connected the capacity analysis to the entropy rate of a hidden Markov chain

Extended prior results to more general input constraints

Abstract

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.