Williamson matrix existence problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\geq1$, an $n\times n$ matrix $A$ is circulant when $A_{ij}=A_{0,(j-i)\bmod n}$ for $0\leq i,j<n$. A Williamson quadruple of order $n$ consists of symmetric circulant matrices $A,B,C,D\in\{1,-1\}^{n\times n}$ satisfying \[A^2+B^2+C^2+D^2=4nI_n. \] Circulant matrices commute. Such a quadruple gives a Hadamard matrix of order $4n$, namely a matrix with entries $\pm1$ and pairwise orthogonal rows, via \[\begin{pmatrix}A&B&C&D\\-B&A&-D&C\\-C&D&A&-B\\-D&-C&B&A\end{pmatrix}. \]

Statement recorded in the supplied source.

Determine the set of all positive integers $n$ for which a Williamson quadruple of order $n$ exists. [S2]

Short English statement

Classify every order admitting a Williamson construction of a Hadamard matrix.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1600014 (AMR-015-0014). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 C. Bright, I. S. Kotsireas and V. Ganesh, Applying Computer Algebra Systems with SAT Solvers to the Williamson Conjecture, arXiv:1804.01172, §2 and §2.1, Theorem 2 and Definition 3. https://arxiv.org/abs/1804.01172.