A New Upper Bound on the Binary Deletion Channel Capacity

Özgür Soysal

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arXiv · 2609.13351

A New Upper Bound on the Binary Deletion Channel Capacity

Özgür Soysal

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Abstract

We prove that the capacity of the binary deletion channel satisfies $C(d)\le (1-d)/4$ for every $13/20\le d<1$. The proof describes the output from right to left, using a six-bit context to assign a description length. We bound the increase in expected description length minus output entropy when one input bit is added. A relative-entropy identity reduces this bound to finitely many linear inequalities. A potential on input windows of length 26 makes the inequalities telescope, giving a bound for every input word. Deletion composition extends the result from $d=13/20$ to all larger deletion probabilities. We also obtain finite-block bounds on mutual information and decoding error, with explicit $O(\log n/n)$ corrections. The finite certificate is checked using exact integer arithmetic, and the proof is formalized end to end in Lean.

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Özgür Soysal. 2026-09-15. A New Upper Bound on the Binary Deletion Channel Capacity. https://arxiv.org/abs/2609.13351

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