Orbit Reduction and Learned Run Distributions for Finite-Blocklength Binary Deletion Channels

Abstract: For a binary deletion channel operating on fixed-length inputs, the relevant figure of merit is the finite-blocklength capacity $C_N(d)=\max_{p(x<sup>N)}\mathrm{I}(X<sup>N;Y)$, not only the infinite-blocklength limit $C(d)$. We show that an optimal input may be chosen constant on complement and permutation-equivalence orbits, reducing the optimization to one weight per orbit, and introduce the optimized run distribution (ORD), an $N$-parameter run-count model that coincides with $C_N(d)$ for $N\le 3$ and is optimal among all run-count-constant inputs. An exact embedding-count dynamic program evaluates $\mathrm{I}$ for these structured laws. A hybrid neural--exact procedure recovers certified ORD weights for $N\le 8$; variational critics (InfoNCE, NWJ, DV/MINE, SMILE) are used only as inner search objectives. Direct score-function learning of ORD weights collapses toward a flat run-length distribution (RLD) for $N\ge 32$. We therefore introduce ORD continuum transfer: a normalized run-count profile learned from exact small-$N$ ORD solutions is resampled at target lengths up to $N=512$. Transferred ORD consistently outperforms RLD at moderate deletion probabilities; at $d=0.1$ the gain vanishes by roughly $N=128$--$192$. Exact ORD rates converted by Fertonani--Duman's length-entropy inequality are valid lower bounds on $C(d)$; nested-Monte-Carlo evaluations of large-$N$ inputs are reported as diagnostics and are not claimed as capacity lower bounds. A fixed-$N=100$ sample-budget study quantifies nested-MC bias. All primary reported rates are values of $\mathrm{I}(X<sup>N;Y)/N$.