Recent progress in the construction of Hadamard matrices

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 $95$ orders up to $10000$ 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.