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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 pairof successive $1$s be separated by at least $d$ $0$s. We consider the problemof designing explicit codes for such channels. In particular, we work with theReed-Muller (RM) family of codes, which were shown by Reeves and Pfister (2021)to achieve the capacity of any unconstrained BMS channel, under bit-MAPdecoding. We show that it is possible to pick $(d,\infty)$-RLL subcodes of acapacity-achieving (over the unconstrained BMS channel) sequence of RM codessuch 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 capacityof the BMS channel. Finally, we also introduce techniques for upper boundingthe rate of any $(1,\infty)$-RLL subcode of a specific capacity-achievingsequence of RM codes.

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