  In this work, we address the question of the largest rate of linear subcodes
of Reed-Muller (RM) codes, all of whose codewords respect a runlength-limited
(RLL) constraint. Our interest is in the $(d,\infty)$-RLL constraint, which
mandates that every pair of successive $1$s be separated by at least $d$ $0$s.
Consider any sequence $\{{\mathcal{C}_m}\}_{m\geq 1}$ of RM codes with
increasing blocklength, whose rates approach $R$, in the limit as the
blocklength goes to infinity. We show that for any linear $(d,\infty)$-RLL
subcode, $\hat{\mathcal{C}}_m$, of the code $\mathcal{C}_m$, it holds that the
rate of $\hat{\mathcal{C}}_m$ is at most $\frac{R}{d+1}$, in the limit as the
blocklength goes to infinity. We also consider scenarios where the coordinates
of the RM codes are not ordered according to the standard lexicographic
ordering, and derive rate upper bounds for linear $(d,\infty)$-RLL subcodes, in
those cases as well. Next, for the setting of a $(d,\infty)$-RLL
input-constrained binary memoryless symmetric (BMS) channel, we devise a new
coding scheme, based on cosets of RM codes. Again, in the limit of blocklength
going to infinity, this code outperforms any linear subcode of an RM code, in
terms of rate, for low noise regimes of the channel.
