A search for Hadamard matrices of Williamson type
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917476489494528 |
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| author | Kharaghani, Hadi Mohammadian, Ali Tayfeh-Rezaie, Behruz |
| author_facet | Kharaghani, Hadi Mohammadian, Ali Tayfeh-Rezaie, Behruz |
| contents | 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 $118$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_08661 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A search for Hadamard matrices of Williamson type Kharaghani, Hadi Mohammadian, Ali Tayfeh-Rezaie, Behruz Combinatorics 05B20, 15B34 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 $118$. |
| title | A search for Hadamard matrices of Williamson type |
| topic | Combinatorics 05B20, 15B34 |
| url | https://arxiv.org/abs/2605.08661 |