Exact asymptotic rate of Delsarte’s linear programming bound for binary and spherical codes found, with key steps from Claude Opus 5.5 and GPT-5.6 Sol
Awaiting review
Importance: major (4 of 5)The takeaway
A coding-theory question open since the 1970s, what Delsarte’s LP bound converges to, now has a closed answer in elliptic integrals; the author credits Opus 5.5 with the step that made the lower bound work.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 16 October 2026
Your AI and this story
- GPT-6 Astra168 days after its cutoff
- Claude Opus 5.5107 days after its cutoff
- Gemini 4 Argon198 days after its cutoff
- Grok 4.7137 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 107 days before it.
Key facts
- Paper: ‘The rate of the Delsarte bound for binary codes and spheres’, Andrew Salmon, arXiv 2610.21263 [math.CO], dated 13 Oct, listed 15 Oct 2026
- Main theorem: for 0 < δ < 1/2 and 0 < s < 1 the lim inf and lim sup of the normalised Delsarte LP optima coincide and equal explicit rate functions J(δ) (binary) and F_S(s) (spherical), parametrised by complete elliptic integrals K, E
- Background: the MRRW (1977) and Kabatiansky–Levenshtein (1978) bounds come from feasible solutions of the Delsarte LP and were the best asymptotic upper bounds for decades; OpenAI’s ‘Ten advances’ Chapter 2 (Aug 2026) improved both; whether the LP itself could do better, and by how much, was open (Kalai 2024, Questions 3–4; Barg–Jaffe 2001 numerics; Samorodnitsky, Navon–Samorodnitsky lower bounds)
- Builds on: Salmon’s earlier ‘The half-rate linear programming bound for binary codes is 1/2 − 1/π’ (arXiv 2609.03736), and OpenAI’s Chapter 1 proof of the Afkhami-Jeddi–Cohn–Hartman–de Laat–Tajdini conjecture for the Cohn–Elkies bound
- Method: lower bounds via potential theory, an obstacle problem solved in closed form; upper bounds from interlacing spectra under the Markov–Krein correspondence; the two match
- AI statement (quoted): ‘The author and GPT-5.6 Sol formulated the upper bounds for spheres and binary codes during the writing of [Sal26] … The author initially worked with Sol and Astra models on incremental strengthenings of the Navon–Samorodnitsky argument … Opus 5.5 identified the compatibility of the MacWilliams transform with the predicted relation … From there, Opus, with light direction from the author, recast the half-rate result as a thin obstacle problem … and in doing so identified the right generalization of [Sal26, Lemma 3.3]. The draft was prepared with Codex.’
What happened
Delsarte’s 1973 linear program gives the best known general upper bounds on the size of binary codes and spherical codes. For fifty years the asymptotic bounds actually used, MRRW (1977) and Kabatiansky–Levenshtein (1978), came from particular feasible solutions of that program, and nobody knew what the program’s true limit was. OpenAI’s August 2026 “Ten advances” improved both bounds; Gil Kalai had asked in 2024 for the limit itself.
On 15 Oct 2026 Andrew Salmon posted arXiv 2610.212631, which computes it: for binary codes at every relative distance below 1/2 and spherical codes at every angle, the normalised Delsarte optimum converges, and the limit is an explicit function of complete elliptic integrals. Lower bounds come from an obstacle problem in potential theory, upper bounds from interlacing spectra of N-qubit constructions related to OpenAI’s Chapter 2, and the two match. It extends his September paper, which found the binary rate at the special distance 1/2 − 1/π.
The AI section is unusually specific. Salmon and GPT-5.6 Sol formulated the upper bounds; work with “Sol and Astra models” on Navon–Samorodnitsky-style arguments improved parts but did not finish; the human realisation of what had to be proved came “on a human timescale”; then “Opus 5.5 identified the compatibility of the MacWilliams transform with the predicted relation”, and “with light direction from the author” recast the problem as a thin obstacle problem and found the right generalisation of his earlier lemma. “The draft was prepared with Codex.”
No Hacker News thread or press coverage was found on 16 Oct (Algolia); X was not searched.
Why it matters
It closes a classical question about the strongest general method for bounding codes, which also matters for sphere packing. It is also a case where models from two labs contributed distinct ideas, recorded step by step by the author.
What is disputed or not yet verified
| Verification | Unrefereed preprint; no formal proof |
|---|
Sources
4 sources from 3 sites. Numbers match the chips in the text.
4 sources: 3 primary, 1 reaction
Primary
- arXiv 2610.21263: The rate of the Delsarte bound for binary codes and spheresarxiv.org, paper
- arXiv 2609.03736: The half-rate linear programming bound for binary codes is 1/2 − 1/π (Salmon)arxiv.org, paper
- OpenAI: Ten advances in mathematics and theoretical computer science (PDF)cdn.openai.com, paper
Reactions
- Gil Kalai: On the limit of the linear programming bound for codes and packing (2024)gilkalai.wordpress.com, discussion
Changes
- Filed from the arXiv PDF