Bounds on the Feedback Capacity of the ($d, \infty$)-RLL Input-Constrained Binary Erasure Channel

The paper considers the input-constrained binary erasure channel (BEC) with causal, noiseless feedback. The channel input sequence respects the (
$d, \infty$
)-runlength limited (RLL) constraint, i.e., any pair of successive 1s must be separated by at least
$d$
0s. We derive upper and lower bounds on the feedback capacity of this channel, given by single parameter maximization problems that differ exclusively in the domain of maximization. The results of Sabag et al. (2016) show that our bounds are tight for the case when
$d=1$
. For the case when
$d=2$
, our lower bound implies that the feedback capacity is equal to the capacity with non-causal knowledge of erasures, for
$\epsilon\in [0,1-\frac{1}{2\log_{2}(3/2)}]$
. The approach in this paper follows Sabag et al. (2017), by deriving single-letter bounds on the feedback capacity, based on output distributions supported on a finite
$Q$
-graph, which is a directed graph with edges labelled by output symbols.