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

TL;DR

This paper introduces a simple feedback coding scheme for the input-constrained binary erasure channel with a runlength limit, achieving the channel's feedback capacity and providing explicit capacity characterization.

Contribution

It presents a novel, zero-error feedback coding scheme for the $(d, Infty)$-RLL constrained BEC, proving it is capacity-achieving and explicitly characterizing the feedback capacity.

Findings

The coding scheme achieves feedback capacity.

Explicit feedback capacity characterization obtained.

Gap identified between feedback and non-feedback capacities for certain constraints.

Abstract

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 $d=1,2$.