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The input-constrained binary erasure channel (BEC) with strictly causalfeedback 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 erasureprobability, $\overline{\varepsilon}=1-\varepsilon$ and $H_2(\cdot)$ is thebinary entropy function. The maximization is only over $\delta_{k-1}$, whilethe 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 forall $k\geq1$. It is shown that the feedback capacity can be achieved usingzero-error, variable length coding. For the converse, an upper bound on thenon-causal setting, where the erasure is available to the encoder just prior tothe transmission, is derived. This upper bound coincides with the lower boundand concludes the search for both the feedback capacity and the non-causalcapacity. As a result, non-causal knowledge of the erasures at the encoder doesnot increase the feedback capacity for the $(0,k)$-RLL input-constrained BEC.This property does not hold in general: the $(2,\infty)$-RLL input-constrainedBEC, where every `$1$' is followed by at least two `$0$'s, is used to show thatthe feedback capacity can be strictly greater than the non-causal capacity.

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