Year: 2026
Type: preprint
Abstract: A Hadamard matrix is a square matrix with entries $\pm 1$ whose rows are mutually orthogonal. Hadamard matrices are conjectured to exist at every order divisible by $4$; for approximately the last twenty years, the smallest open case of this conjecture was order 668. In August 2026, Alpöge, Voinov and Reynolds-Haertle announced Hadamard matrices at the twelve orders below $2000$ at which none was known. In this paper we verify and analyse the claimed matrices. We survey results on the existence of Hadamard matrices and list the $85$ orders up to $10\,000$ at which no Hadamard matrix is currently known. In pursuing this work we constructed Hadamard matrices at sixteen orders that were open in the published literature. We conclude with a list of open problems.
Source: arXiv (Cornell University)
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223155683