Coding Schemes Based on Reed-Muller Codes for (d,∞)-RLL Input-Constrained Channels
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The paper considers coding schemes derived from Reed-Muller (RM) codes, for transmission over input-constrained memoryless channels. Our focus is on the (d,∞)-runlength limited (RLL) constraint, which mandates that any pair of successive 1s be separated by at least d 0s. In our study, we first consider (d,∞)-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,∞)-RLL subcodes of rate R·2^-⌈log_2(d+1)⌉. We then show that our construction is essentially rate-optimal, by deriving an upper bound on the rates of linear (d,∞)-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,∞)-RLL subcodes that achieve a rate of max(0,R-3/8). This, for R > 3/4, beats the R/2 rate obtainable from linear subcodes. We further derive upper bounds on the rates of (1,∞)-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,∞)-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,∞)-RLL subcodes, for values of R close to 1.
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