# 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≥ 1, and is given by C^fb_(0,k)(ε) = εH_2(δ_0)+∑_i=1^k-1(ε^i+1H_2(δ_i)∏_m=0^i-1δ_m)/1+∑_i=0^k-1(ε^i+1∏_m=0^iδ_m), where ε is the erasure probability, ε=1-ε and H_2(·) is the binary entropy function. The maximization is only over δ_k-1, while the parameters δ_i for i≤ k-2 are straightforward functions of δ_k-1. The lower bound is obtained by constructing a simple coding for all k≥1. 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,∞)-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.

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