On the Performance of Reed-Muller Codes Over (d,∞)-RLL Input-Constrained BMS Channels
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This paper considers the input-constrained binary memoryless symmetric (BMS) channel, without feedback. The channel input sequence respects the (d,∞)-runlength limited (RLL) constraint, which mandates that any pair of successive 1s be separated by at least d 0s. 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,∞)-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·2^-⌈log_2(d+1)⌉, where C is the capacity of the BMS channel. Finally, we also introduce techniques for upper bounding the rate of any (1,∞)-RLL subcode of a specific capacity-achieving sequence of RM codes.
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