  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.
