Numerically Computable Lower Bounds on the Capacity of the (1, ∞)-RLL Input-Constrained Binary Erasure Channel

Abstract

The paper considers the binary erasure channel (BEC) with the inputs to the channel obeying the (1, ∞)-runlength limited (RLL) constraint, which forbids input sequences with consecutive ones. We derive a lower bound on the capacity of the channel, by considering the mutual information rate between the inputs and the outputs when the input distribution is first-order Markov. Further, we present a numerical algorithm for numerically computing the lower bound derived. The algorithm is based on ideas from stochastic approximation theory, and falls under the category of two-timescale stochastic approximation algorithms. We provide numerical evaluations of the lower bound, and characterize the input distribution that achieves the bound. We observe that our numerical results align with those obtained using the sampling-based scheme of Arnold et al. (2006). Furthermore, we note that our lower bound expression recovers the series expansion type lower bound discussed in Corollary 5 of Li and Han (2018). We also derive an alternative single-parameter optimization problem as a lower bound on the capacity, and demonstrate that this new bound is better than the linear lower bound shown in Li and Han (2018) and Rameshwar and Kashyap (2020), for ∊ $\\gtrapprox$ 0.77, where ∊ is the erasure probability of the channel.