On the Performance of Reed-Muller Codes Over (d, ∞)-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, ∞)-runlength limited (RLL) constraint, which mandates that any pair of successive 1s be separated by at least d 0s. 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, ∞)-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, ∞)-RLL subcode of a specific capacity-achieving sequence of RM codes.