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.
翻译:本文审议了输入限制的二进制对称( BMS) 频道, 没有反馈。 频道输入序列尊重 $( d,\ infty) $- run- under 限值 (RLL) 限制, 要求任何连续的一对$( infty) $- RLL 子代码至少以美元分隔。 我们考虑为这些频道设计明确的代码的问题。 特别是, 我们与 Reed- Muller (RM) 的代码组合作, Reed- Muller (RM) 显示于 Reeves 和 Pfister (2021), 以在 Bit- MAP 解码下实现任何未受限制的 BMS 频道 频道的能力 2( d+1)\ rightrcyel 频道的能力。 我们显示, $C 的 RLULL 子代码可以选择一个能力( 未受限制的 BMS 频道 ) 具体 RB1 的 RB1 的 RBC\\\\ recl) 程序约束能力 。