On the Performance of Reed-Muller Codes Over $(d,\infty)$-RLL Input-Constrained BMS Channels

Abstract

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.

Linear Runlength-Limited Subcodes of Reed-Muller Codes and Coding Schemes for Input-Constrained BMS Channels

“…Note that C (𝑑,∞) 𝑚 (𝑅) is a linear subcode of C 𝑚 (𝑅). The following theorem from [9] then holds: [9]). For any 𝑅 ∈ (0, 𝐶), the sequence of linear codes {C (𝑑,∞)…”

“…The results summarized in the previous sections provide lower and upper bounds on achievable rates by using subcodes of RM codes. In particular, Theorem III.3 (Theorem III.2 of [9]) shows that, using subcodes of RM codes, rates of up to 2 − log 2 (𝑑+1) • 𝐶 are achievable over (𝑑, ∞)-RLL input-constrained BMS channels. In this section, we provide another construction, which uses cosets of RM codes.…”

“…The coding scheme in this paper achieves rates close to the lower bound in [10], for values of 𝐶 close to 1. We note that using linear (𝑑, ∞)-RLL subcodes of RM codes as in [9], we can achieve larger rates as compared to the rate lower bound in [10], when the capacity 𝐶 is low, i.e., when…”

“…In this paper, we prove that any linear RM subcode that respects the (𝑑, ∞)-RLL constraint, must have a rate of at most 𝐶 𝑑+1 , in the limit as the blocklength goes to infinity. In doing so, we show that one cannot do better, asymptotically, than the simple coding scheme in [9], if one requires that the subcodes be linear. We also consider the rates achieved using linear (𝑑, ∞)-RLL subcodes of permuted RM codes, and show that for codes of large enough blocklength, almost all permutations must respect an upper bound of 𝐶 𝑑+1 + 𝛿, for 𝛿 being as small as is required.…”

“…Suppose that 𝐶 is the capacity of the unconstrained channel. The authors in [9] showed a simple linear coding scheme, using subcodes of RM codes, with rates of 𝐶 • 2 − log 2 (𝑑+1) , being achievable over (𝑑, ∞)-RLL inputconstrained BMS channels. In this paper, we prove that any linear RM subcode that respects the (𝑑, ∞)-RLL constraint, must have a rate of at most 𝐶 𝑑+1 , in the limit as the blocklength goes to infinity.…”