# The Feedback Capacity of the Binary Erasure Channel With a No-Consecutive-Ones Input Constraint Academic Article

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### abstract

• The input-constrained erasure channel with feedback is considered, where the binary input sequence contains no consecutive ones, i.e., it satisfies the $(1,\infty )$ -RLL constraint. We derive the capacity for this setting, which can be expressed as $C_{\epsilon }=\max _{0\leq p\leq 0.5} \frac {(1-\epsilon )H_{b}(p)}{1+(1-\epsilon )p}$ , where $\epsilon$ is the erasure probability and $H_{b}(\cdot )$ is the binary entropy function. Moreover, we prove that a priori knowledge of the erasure at the encoder does not increase the feedback capacity. The feedback capacity was calculated using an equivalent dynamic programming (DP) formulation with an optimal average-reward that is equal to the capacity. Furthermore, we obtained an optimal encoding procedure from the solution of the DP, leading to a capacity-achieving, zero-error coding scheme for our setting. DP is, thus, shown to be a tool not only for solving optimization problems, such as capacity calculation, but also for constructing optimal coding schemes. The derived capacity expression also serves as the only non-trivial upper bound known on the capacity of the input-constrained erasure channel without feedback, a problem that is still open.

### publication date

• January 1, 2016