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arXiv · 2605.08661

A search for Hadamard matrices of Williamson type

Hadi Kharaghani·Ali Mohammadian·Behruz Tayfeh-Rezaie
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Abstract

In this article, we consider a special class of Williamson type matrices which we call them near Williamson matrices. They are in fact four $n\times n$ $(-1, 1)$-matrices $A, B, C, D$ so that $A$ is circulant, $B,C,D$ are symmetric circulant, and they satisfy $AA^\top+BB^\top+CC^\top+DD^\top=4nI$. Using a computer search, we find all inequivalent near Williamson matrices for all odd orders at most $35$. We also show that such matrices exist for all odd orders up to $63$. As a consequence, we find the first known example of a quaternary Hadamard matrix of order $118$.

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Hadi Kharaghani, Ali Mohammadian, Behruz Tayfeh-Rezaie. 2026-05-09. A search for Hadamard matrices of Williamson type. https://arxiv.org/abs/2605.08661

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