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 zeros are allowed. The feedback capacity of this channel is derived for all k≥1 C^fn_{(0, k)}(ϵ) = max ϵH₂(δ₀)+Σ^k-1_i=1(ϵⁱ⁺¹H₂)(δ_i)Π^i-1_m=0δ_m)/1+Σ^k-1_i=0(ϵⁱ⁺¹Πⁱ_m=0δ_m) where ϵ is the erasure probability, ϵ=1-ϵ, H₂(·) is the binary entropy function and the maximization is only over δ_k-1, while the other parameters δ₀,..., δ_k-2 are simple functions of δ_k-1. A simple coding scheme is constructed for all k, establishing that the feedback capacity can be achieved using variable length zero-error coding. In addition, it is shown that non-causal knowledge of the erasures at the encoder does not increase the feedback capacity.