Claude-assisted constructions complete Hadamard matrices for every order below 2000
Including 668
Result confirmed
The takeaway
Claude-assisted searches constructed Hadamard matrices for the 12 remaining unknown orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964).
Status
- Claim
Result confirmed
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 21 October 2026
Your AI and this story
- GPT-6 Astra104 days after its cutoff
- Claude Opus 5.543 days after its cutoff
- Gemini 4 Hydrogen134 days after its cutoff
- Grok 4.773 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 43 days before it.
Key facts
- Orders constructed: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964
- Order 668 was the smallest unknown order since 428 was constructed in 2005
- People: Levent Alpöge, P. Voinov, S. Reynolds-Haertle; order 668 was an Epoch AI ‘open problem’ entry
- Independent verification (Egan, Kotsireas, London, Ó Catháin; arXiv 2610.27175, 19 Oct 2026): they ‘verify and analyse the claimed matrices’; each is a Goethals–Seidel array over an abelian group (nine over cyclic groups, three over non-cyclic groups); ‘A Hadamard matrix is therefore known at every order below 2000 divisible by 4’
- The same paper builds Hadamard matrices at sixteen further orders below 10,000 that were open in the published literature (e.g. 2076 via a symmetric BH(86, 4), 3732, 4492, 5020, 5932, 5788, 8476), and lists 95 orders ≤ 10,000 still open, the smallest 2396 = 4·599
- Its AI statement: ‘The authors used Anthropic’s Claude (Claude Opus 5.5, via Claude Code, September–October 2026) extensively in this work. Claude wrote the Python and C++ programs used to analyse the twelve matrices … and to construct the symmetric BH(86, 4), the T-matrices of orders 151 and 163, and the Hadamard matrices of orders 5788 and 8476’; every matrix is public at math74.com/Hadamard and ‘verified by an exact computation of HH^T’
What happened
Guided searches built the missing matrices, verifiable by simple matrix multiplication.
Why it matters
It closed a famous “smallest unknown case” that had stood for two decades, with an easily verified result.
Sources
3 sources from 3 sites. Numbers match the chips in the text.
3 sources: 2 primary, 1 reaction
Primary
- Epoch AI open problems: Hadamard matrix of order 668epoch.ai, official
- arXiv 2610.27175: Recent progress in the construction of Hadamard matrices (Egan, Kotsireas, London, Ó Catháin)arxiv.org, paper
Reactions
- John D. Cook: Constructing Hadamard matricesjohndcook.com, discussion
Changes
- Added the independent verification and 16 new orders by Egan, Kotsireas, London and Ó Catháin (arXiv 2610.27175, Claude Opus 5.5 used extensively)
- Filed