An optimal coding scheme for the BIBO channel with a no-repeated-ones input constraint

A binary-input binary-output (BIBO) channel is investigated in the presence of feedback and input constraints. The feedback capacity and the optimal input distribution of this setting are presented for the case where the input sequence contains no consecutive ones. A simple coding scheme is designed based on the principle of posterior matching, which was introduced by Shayevitz and Feder for memoryless channels. The posterior matching scheme for our input-constrained setting is shown to achieve capacity using two new ideas: message history, which captures the memory embedded in the setting, and message splitting, which simplifies the scheme analysis. Additionally, in the special case of an S-channel, we give a very simple zero-error coding scheme that achieves capacity.