Determine exact large-blocklength ORD solutions

Determine the exact optimized run distribution (ORD) solutions for large blocklengths of the binary deletion channel, where direct optimization is currently infeasible, in order to assess whether normalized small-blocklength ORD geometry accurately transfers to those blocklengths.

Background

The paper optimizes input distributions within the optimized run distribution (ORD) family, whose weights assign probability to binary strings according to their run count. Exact ORD solutions can be computed for small blocklengths using the embedding-count dynamic program, but direct optimization becomes impractical at larger blocklengths. The authors therefore transfer a normalized run-count profile learned from small blocklengths to larger targets and compare it with the flat run-length distribution baseline. Exact large-blocklength ORD solutions would provide the missing benchmark for determining how accurately this transfer procedure captures the true optimum.

References

The objective is not to reproduce exact large-$N$ ORD solutions, which are unknown, but to determine whether transferred run structure continues to outperform the flat RLD baseline---and, if so, over what range of blocklengths.

— Orbit Reduction and Learned Run Distributions for Finite-Blocklength Binary Deletion Channels  (2609.24889 - Khodaiemehr et al., 21 Sep 2026) in Section 4, “Motivation and problem formulation” (Section \ref{sec:transfer})

A remaining theoretical question is whether maximizers of~eq:dm-lb remain tight as $Z\to\infty$ without a hard cap, especially as $d\to1$.

— Improved Lower Bounds on the Capacity of the Binary Deletion Channel via a Learning Approach to Run-Length Inputs  (2609.24908 - Khodaiemehr et al., 21 Sep 2026) in Section “Toward stronger run-length bounds,” paragraph “Support growth and convergence” (Subsection \ref{subsec:rld-future})

At present the run-length constellation phenomenon is purely empirical. Several theoretical questions naturally arise:

\begin{enumerate}

\item Do maximizers of $\mathcal R(P,d)$ eventually suppress short runs as $d\rightarrow 1$?

\item Must every maximizing sequence of distributions become multimodal at sufficiently large deletion probabilities?

\item Is the approximately constant spacing observed in units of $\sigma(z)$ an asymptotic property of the optimizer?

\item Can one characterize maximizing laws through a finite set of active support points satisfying first-order optimality conditions?

\end{enumerate}

— Improved Lower Bounds on the Capacity of the Binary Deletion Channel via a Learning Approach to Run-Length Inputs  (2609.24908 - Khodaiemehr et al., 21 Sep 2026) in Section “Toward stronger run-length bounds,” paragraph “Toward analytical structure theorems” (Subsection \ref{subsec:rld-future})

Whether analogous run-length optimization and certification techniques can yield improved insertion-channel bounds remains an open question.

— Improved Lower Bounds on the Capacity of the Binary Deletion Channel via a Learning Approach to Run-Length Inputs  (2609.24908 - Khodaiemehr et al., 21 Sep 2026) in Section “Discussion and Conclusions,” final paragraph