{"id":"openai-math/The-circulant-Hadamard-conjecture","pdf":"https://github.com/openai/math/blob/main/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-circulant-Hadamard-conjecture-September-23-2026","tags":["combinatorics","hadamard-matrices","barker-sequences"],"score":0,"title":"The circulant Hadamard conjecture","domain":"Combinatorics","family":"179","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":true,"family_title":"The circulant Hadamard and Barker-sequence conjectures"}