  The paper considers coding schemes derived from Reed-Muller (RM) codes, for
transmission over input-constrained memoryless channels. Our focus is on the
$(d,\infty)$-runlength limited (RLL) constraint, which mandates that any pair
of successive $1$s be separated by at least $d$ $0$s. In our study, we first
consider $(d,\infty)$-RLL subcodes of RM codes, taking the coordinates of the
RM codes to be in the standard lexicographic ordering. We show, via a simple
construction, that RM codes of rate $R$ have linear $(d,\infty)$-RLL subcodes
of rate $R\cdot{2^{-\left \lceil \log_2(d+1)\right \rceil}}$. We then show that
our construction is essentially rate-optimal, by deriving an upper bound on the
rates of linear $(d,\infty)$-RLL subcodes of RM codes of rate $R$. Next, for
the special case when $d=1$, we prove the existence of potentially non-linear
$(1,\infty)$-RLL subcodes that achieve a rate of
$\max\left(0,R-\frac38\right)$. This, for $R > 3/4$, beats the $R/2$ rate
obtainable from linear subcodes. We further derive upper bounds on the rates of
$(1,\infty)$-RLL subcodes, not necessarily linear, of a certain canonical
sequence of RM codes of rate $R$. We then shift our attention to settings where
the coordinates of the RM code are not ordered according to the lexicographic
ordering, and derive rate upper bounds for linear $(d,\infty)$-RLL subcodes in
these cases as well. Finally, we present a new two-stage constrained coding
scheme, again using RM codes of rate $R$, which outperforms any linear coding
scheme using $(d,\infty)$-RLL subcodes, for values of $R$ close to $1$.
