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.
翻译:在这项工作中,我们处理的是Reed-Muller(RM)代码线性子代码的最大比例值问题,所有代码都尊重限值限制的限值限制。我们感兴趣的是$(d),\infty)$-RLL(RL)限制,这要求每对连续一美元的代码至少分离美元美元。考虑到任何序列,美元=mathcal{C ⁇ m\geq 1美元,轮长增加,费率接近R$(美元),在轮长到不精确的限度内限内。我们还要研究对于任何线性美元(d,infty)$-RLLL(美元)子代码,$\hhat\mathcal{C%m(美元),这要求每对连续一对一对一对一美元,每对每对每对每对每对一的一一美元, 美元,每对每对每对每对每对每对每对每对每对一美元,至少美元,每对每对每对每对每对每对每对一美元, 美元, 美元,每对每对每对每对每对每对每对每对每对每对的,每对,每对,每对,每对每对每对每对,每对每对每对每对每对每对一个,每对每对,每对,每对,每对每对,每对,每对,每对每对每对每对每对,每对,每对,每对一个, 美元, 美元, 美元,每对,每对,每整整整整一个,每对一个,每整一个,每对一次,每对一个,其,其,每对一个,每对一个,每对一个,每对一次,每对一个,每对一次,每对一次,每对一个,每对一个,每对一个,每对一个,每对一个,每对一个,每对一个,每对一个,每对一个,每对一个,每对一个,每对一个,每计算一个,每计算一个,每计算一个,每计算一个,每计算一个,每计算一个,每递一次,每计算的计算的计算的,每计算的计算的计算的计算的计算的计算的,每