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Circulant Hadamard Conjecture (Nonexistence beyond order 4)

Prove that for any positive integer N, the only circulant real Hadamard matrices H ∈ M_N(±1) are the order-4 matrix K_4 and its conjugates.

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Background

Circulant matrices have entries determined by a single vector via M_{ij} = ξ_{j−i}. One can ask whether such structured matrices can be Hadamard. For N=4, explicit circulant Hadamard matrices (K_4 and conjugates) exist, but for larger N no examples are known. The Circulant Hadamard Conjecture posits that these order-4 examples are the only circulant Hadamard matrices, a problem raised by Ryser and long open in combinatorics.

References

\begin{conjecture}[Circulant Hadamard Conjecture (CHC)] The only Hadamard matrices which are circulant are $$K_4=\begin{pmatrix}-1&1&1&1\ 1&-1&1&1\ 1&1&-1&1\ 1&1&1&-1\end{pmatrix}$$ and its conjugates, regardless of the value of $N\in\mathbb N$. \end{conjecture}

— Advanced linear algebra (2506.18666 - Banica, 23 Jun 2025) in Chapter 11b. Hadamard matrices