The feedback capacity of the binary symmetric channel with a no-consecutive-ones input constraint

The binary symmetric channel (BSC) with feedback is considered, where the input sequence contains no consecutive ones, i.e., satisfies the (1,∞)-RLL constraint. In [1], the capacity of this setting was formulated as dynamic programming (DP); however, analytic expressions for capacity and optimal input distribution were left as an open problem. In this paper, we derive explicit expressions for both feedback capacity and optimal input distribution. The solution was obtained by using an equivalent DP and solving its corresponding Bellman equation. The feedback capacity also serves as an upper bound on the capacity of the input-constrained BSC channel without feedback, a problem that is still open.