---
id: "2026-08-12-claude-hadamard-matrices-below-2000"
url: "https://postcutoff.com/e/2026-08-12-claude-hadamard-matrices-below-2000/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-08-12"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Result confirmed]
verification: Preprint only
sources: 3
editor: human editor
human_review: null
version: "2026-10-21"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-08-12-claude-hadamard-matrices-below-2000/

# Claude-assisted constructions complete Hadamard matrices for every order below 2000

Full title: Claude-assisted constructions complete Hadamard matrices for every order below 2000, including 668

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). Order 668 had been the smallest open case of the Hadamard conjecture for about 21 years.

## 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.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 104 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 43 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 134 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 73 days after its cutoff

## Sources

1. [Epoch AI open problems: Hadamard matrix of order 668](https://epoch.ai/frontiermath/open-problems/hadamard) (epoch.ai, official)
2. [arXiv 2610.27175: Recent progress in the construction of Hadamard matrices (Egan, Kotsireas, London, Ó Catháin)](https://arxiv.org/abs/2610.27175) (arxiv.org, paper)
3. [John D. Cook: Constructing Hadamard matrices](https://www.johndcook.com/blog/2026/08/13/constructing-hadamard-matrices/) (johndcook.com, discussion)

## Changes

- 2026-10-21: Added the independent verification and 16 new orders by Egan, Kotsireas, London and Ó Catháin (arXiv 2610.27175, Claude Opus 5.5 used extensively)
- 2026-09-29 (filed): Created

## Related

- 2026-08-23: [Claude-assisted search breaks the elliptic curve rank record](https://postcutoff.com/e/2026-08-23-elliptic-curve-rank-record-31/index.md)
- 2026-08-10: [Claude proves more than two-thirds of Riemann zeta zeros are simple and on the critical line (up from 41.6%)](https://postcutoff.com/e/2026-08-10-claude-riemann-zeta-zeros-two-thirds/index.md)
- People: [Levent Alpöge](https://postcutoff.com/person/levent-alpoge/)