On the Performance of Reed-Muller Codes Over $(d,\infty)$-RLL Input-Constrained BMS Channels

This paper considers the input-constrained binary memoryless symmetric (BMS) channel, without feedback. The channel input sequence respects the $(d,\infty)$-runlength limited (RLL) constraint, which mandates that any pair of successive $1$s be separated by at least $d$ $0$s. We consider the problem of designing explicit codes for such channels. In particular, we work with the Reed-Muller (RM) family of codes, which were shown by Reeves and Pfister (2021) to achieve the capacity of any unconstrained BMS channel, under bit-MAP decoding. We show that it is possible to pick $(d,\infty)$-RLL subcodes of a capacity-achieving (over the unconstrained BMS channel) sequence of RM codes such that the subcodes achieve, under bit-MAP decoding, rates of $C\cdot{2^{-\left \lceil \log_2(d+1)\right \rceil}}$, where $C$ is the capacity of the BMS channel. Finally, we also introduce techniques for upper bounding the rate of any $(1,\infty)$-RLL subcode of a specific capacity-achieving sequence of RM codes.