Abstract:
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) -ru...Show MoreMetadata
Abstract:
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 1s be separated by at least d~0\text{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-\frac {3}8}\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.
Published in: IEEE Transactions on Information Theory ( Volume: 69, Issue: 11, November 2023)