Feedback Capacity and Coding for the $(0,k)$-RLL Input-Constrained BEC

The input-constrained binary erasure channel (BEC) with strictly causal feedback is studied. The channel input sequence must satisfy the $(0,k)$-runlength limited (RLL) constraint, i.e., no more than $k$ consecutive `$0$'s are allowed. The feedback capacity of this channel is derived for all $k\geq 1$, and is given by $$C^\mathrm{fb}_{(0,k)}(\varepsilon) = \max\frac{\overline{\varepsilon}H_2(\delta_0)+\sum_{i=1}^{k-1}\left(\overline{\varepsilon}^{i+1}H_2(\delta_i)\prod_{m=0}^{i-1}\delta_m\right)}{1+\sum_{i=0}^{k-1}\left(\overline{\varepsilon}^{i+1} \prod_{m=0}^{i}\delta_m\right)},$$ where $\varepsilon$ is the erasure probability, $\overline{\varepsilon}=1-\varepsilon$ and $H_2(\cdot)$ is the binary entropy function. The maximization is only over $\delta_{k-1}$, while the parameters $\delta_i$ for $i\leq k-2$ are straightforward functions of $\delta_{k-1}$. The lower bound is obtained by constructing a simple coding for all $k\geq1$. It is shown that the feedback capacity can be achieved using zero-error, variable length coding. For the converse, an upper bound on the non-causal setting, where the erasure is available to the encoder just prior to the transmission, is derived. This upper bound coincides with the lower bound and concludes the search for both the feedback capacity and the non-causal capacity. As a result, non-causal knowledge of the erasures at the encoder does not increase the feedback capacity for the $(0,k)$-RLL input-constrained BEC. This property does not hold in general: the $(2,\infty)$-RLL input-constrained BEC, where every `$1$' is followed by at least two `$0$'s, is used to show that the feedback capacity can be strictly greater than the non-causal capacity.