PreprintCC0arXiv
A search for Hadamard matrices of Williamson type
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× n $(-1, 1)$-matrices $A, B, C, D$ so that A is circulant, B,C,D are symmetric circulant, and they satisfy AA⊤+BB⊤+CC⊤+DD⊤=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 $118$.
Related
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BibTeX
@misc{kharaghani2026search,
title = {{A search for Hadamard matrices of Williamson type}},
author = {Hadi Kharaghani and Ali Mohammadian and Behruz Tayfeh-Rezaie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.08661},
eprint = {2605.08661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.08661},
note = {CC0},
}
RIS
TY - UNPB
TI - A search for Hadamard matrices of Williamson type
AU - Hadi Kharaghani
AU - Ali Mohammadian
AU - Behruz Tayfeh-Rezaie
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2605.08661
UR - https://arxiv.org/abs/2605.08661
AB - 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× n $(-1, 1)$-matrices $A, B, C, D$ so that A is circulant, B,C,D are symmetric circulant, and they satisfy AA^⊤+BB^⊤+CC^⊤+DD^⊤=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 $118$.
ER -