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A Feedback Capacity-Achieving Coding Scheme for the $(d,\infty)$-RLL Input-Constrained Binary Erasure Channel

This paper considers the memoryless input-constrained binary erasure channel (BEC). The channel input constraint is the $(d,\infty)$-runlength limited (RLL) constraint, which mandates that any pair of successive $1$s in the input sequence be separated by at least $d$ $0$s. We consider a scenario where there is causal, noiseless feedback from the decoder. We demonstrate a simple, labelling-based, zero-error feedback coding scheme, which we prove to be feedback capacity-achieving, and, as a by-product, obtain an explicit characterization of the feedback capacity. Our proof is based on showing that the rate of our feedback coding scheme equals an upper bound on the feedback capacity derived using the single-letter bounding techniques of Sabag et al. (2017). Further, we note using the tools of Thangaraj (2017) that there is a gap between the feedback and non-feedback capacities of the $(d,\infty)$-RLL input constrained BEC, at least for $d=1,2$.