Coding Schemes Based on Reed-Muller Codes for (d, ∞)-RLL Input-Constrained Channels

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.