Improved Lower Bounds on the Capacity of the Binary Deletion Channel via a Learning Approach to Run-Length Inputs

The capacity $C(d)$ of the i.i.d. binary deletion channel exists by Dobrushin’s information-stability theorem, but no closed form is known. Classical constructive lower bounds from i.i.d. run-length coding have been evaluated only for one- or two-parameter families (geometric, Markov, or Morse-type). We show that the same infinite-blocklength functionals become strictly stronger when the run-length law $P$ is treated as a free distribution and optimized by learning. We reduce the Drinea–Mitzenmacher functional to a bilinear form in $P$ and prove that finite-support truncation is one-sided, so computed values remain valid lower bounds. We extend the Venkataramanan et al. reductions from geometric runs to arbitrary finite-support laws, including a residual-run HMM for output-bit entropy. Softmax gradient ascent searches $P$; every reported number is a fresh one-sided evaluation of the formula, with no Monte Carlo and no finite length-entropy penalty. The envelope of the two optimized bounds exceeds Gallager’s $1-h(d)$ (for $d<1\mathrm{/}2$) and the tabulated bounds of Drinea–Mitzenmacher, Venkataramanan et al., and Rubinstein–Con at every tested $d$. Representative values: $C(d)\ge 0.92212$, $0.72939$, $0.56486$, $0.35127$, $0.22616$, $0.10414$, $0.02891$, $0.01322$ at $d=0.01$, $0.05$, $0.10$, $0.20$, $0.30$, $0.50$, $0.80$, $0.90$. The largest absolute gain over that record is $3.7\times {10}^{-3}$ bits (at $d=0.30$); the largest relative gain is $6.8\mathrm{\%}$ (at $d=0.90$). For $d\le 0.45$ the envelope is the free-$P$ Venkataramanan functional; from $d=0.50$ it is the learned Drinea–Mitzenmacher law. At large $d$ the optimizer finds sparse run-length combs that parametric families cannot represent. A concurrent enclosure of Papailiopoulos is stronger on much of $[0,1]$, but our envelope remains larger at high $d$ (e.g. $0.02891$ vs $0.02884$ at $d=0.80$; $0.01322$ vs $0.01293$ at $d=0.90$).