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 $1$.

翻译：本文研究了源于Reed-Muller（RM）码的编码方案，用于输入约束无记忆信道上的传输。我们聚焦于$(d,\infty)$-游程长度受限（RLL）约束，该约束要求任意两个相邻的$1$之间至少间隔$d$个$0$。研究中，我们首先考虑RM码的$(d,\infty)$-RLL子码，其中RM码的坐标采用标准字典序排列。通过简单构造，我们证明速率为$R$的RM码具有速率为$R\cdot{2^{-\left \lceil \log_2(d+1)\right \rceil}}$的线性$(d,\infty)$-RLL子码。随后，通过推导速率为$R$的RM码中线性$(d,\infty)$-RLL子码速率的上界，我们证明该构造在速率意义上本质上是最优的。其次，针对$d=1$的特殊情况，我们证明了潜在非线性$(1,\infty)$-RLL子码的存在性，其可达速率为$\max\left(0,R-\frac38\right)$。当$R > 3/4$时，该速率优于线性子码可实现的$R/2$速率。我们进一步推导了速率为$R$的某规范RM码序列中（未必线性的）$(1,\infty)$-RLL子码的速率上界。然后，我们将关注点转向RM码坐标不按字典序排列的情形，并同样推导了这种情况下线性$(d,\infty)$-RLL子码的速率上界。最后，我们提出一种新的两阶段约束编码方案，再次使用速率为$R$的RM码，在$R$接近$1$时，该方案优于任何使用$(d,\infty)$-RLL子码的线性编码方案。