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

Abstract

The input-constrained binary erasure channel (BEC) with strictly causal feedback is studied. The channel input sequence must satisfy the <inline-formula> <tex-math notation= LaTeX >$(0,k)$ </tex-math></inline-formula>-runlength limited (RLL) constraint, i.e., no more than <inline-formula> <tex-math notation= LaTeX >$k$ </tex-math></inline-formula> consecutive ‘0’s are allowed. The feedback capacity of this channel is derived for all <inline-formula> <tex-math notation= LaTeX >$k\\geq 1$ </tex-math></inline-formula>, and is given by <inline-formula> <tex-math notation= LaTeX >$C^{\\mathrm {fb}}_{(0,k)}(\\varepsilon ) = \\max \\frac {\\overline {\\varepsilon }H_{2}(\\delta _{0})+\\sum _{i=1}^{k-1}\\left ({\\overline {\\varepsilon }^{i+1}H_{2}(\\delta _{i})\\prod _{m=0}^{i-1}\\delta _{m}}\\right )}{1+\\sum _{i=0}^{k-1}\\left ({\\overline {\\varepsilon }^{i+1} \\prod _{m=0}^{i}\\delta _{m}}\\right )}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation= LaTeX >$\\varepsilon $ </tex-math></inline-formula> is the erasure probability, <inline-formula> <tex-math notation= LaTeX >$\\overline {\\varepsilon }=1-\\varepsilon $ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$H_{2}(\\cdot )$ </tex-math></inline-formula> is the binary entropy function. The maximization is only over <inline-formula> <tex-math notation= LaTeX >$\\delta _{k-1}$ </tex-math></inline-formula>, while the parameters <inline-formula> <tex-math notation= LaTeX >$\\delta _{i}$ </tex-math></inline-formula> for <inline-formula> <tex-math notation= LaTeX >$i\\leq k-2$ </tex-math></inline-formula> are straightforward functions of <inline-formula> <tex-math notation= LaTeX >$\\delta _{k-1}$ </tex-math></inline-formula>. The lower bound is obtained by constructing a simple coding for all <inline-formula> <tex-math notation= LaTeX >$k\\geq 1$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation= LaTeX >$(0,k)$ </tex-math></inline-formula>-RLL input-constrained BEC. This property does not hold in general: the <inline-formula> <tex-math notation= LaTeX >$(2,\\infty )$ </tex-math></inline-formula>-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 smaller than the non-causal capacity.