Smallest unsolved instance of the Hadamard conjecture
Description of constant
A Hadamard matrix of order $n$ is an $n\times n$ matrix $H$ with entries in $\{-1,1\}$ such that
\[HH^{\top} = n I_n.\]Equivalently, the rows (and columns) are pairwise orthogonal.
It is known that Hadamard matrices can exist only for $n=1,2$, or $n\equiv 0 \pmod{4}$.
We define $C_{23}$ to be the smallest integer $n\equiv 0 \pmod{4}$ such that there is no Hadamard matrix of order $n$. If no such $n$ exists, we set $C_{23}=\infty$.
The Hadamard conjecture asserts that $C_{23}=\infty$, i.e. that Hadamard matrices exist for every order $n\equiv 0 \pmod{4}$.
(Equivalently, by Hadamard’s determinant inequality, for $A\in\{-1,1\}^{n\times n}$ one has $|\det(A)|\le n^{n/2}$, with equality iff $A$ is Hadamard; the conjecture predicts equality is attainable for all $n\equiv 0\pmod4$.)
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $\infty$ | Trivial | No finite upper bound is known; conjecturally sharp (Hadamard conjecture). |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $4$ | Trivial | By definition $C_{23}$ (if finite) is a multiple of $4$. |
| $668$ | [CP2024] | All orders $n<668$ with $n\equiv 0\pmod4$ are known to admit Hadamard matrices; the smallest currently unresolved order is $668$. |
Additional comments and links
- A benchmark open instance is the existence of a Hadamard matrix of order $668$; in the range $n\le 1208$, the only unresolved orders are $668,716,892,1132$ [CP2024].
- Classical infinite families of Hadamard orders include:
- Sylvester’s Kronecker-product construction, giving Hadamard matrices of order $2^m$ for every $m\ge 0$ [Syl1867].
- Paley’s constructions, giving Hadamard matrices of order $q+1$ when $q$ is a prime power with $q\equiv 3\pmod{4}$, and of order $2(q+1)$ when $q$ is a prime power with $q\equiv 1\pmod{4}$ [Pal1933].
- Wikipedia page on Hadamard matrices
- SageMath implementation and access to Sloane’s library: SageMath documentation
References
- [CP2024] Cati, Matteo; Pasechnik, Dmitrii V. A database of constructions of Hadamard matrices. arXiv:2411.18897 (2024/2025).
- [Had1893] Hadamard, Jacques. Résolution d’une question relative aux déterminants. Bull. Sci. Math. (2) 17 (1893), 240–246.
- [Syl1867] Sylvester, James J. Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tessellated pavements in two or more colours, with applications to Newton’s rule, ornamental art, and numerous other subjects. Philosophical Magazine 34 (1867), 461–475. DOI: 10.1080/14786446708639914.
- [Pal1933] Paley, Raymond E. A. C. On orthogonal matrices. J. Math. Phys. 12 (1933), 311–320.
- [W1944] Williamson, John. Hadamard’s determinant theorem and the sum of four squares. Duke Math. J. 11 (1944), no. 1, 65–81. DOI: 10.1215/S0012-7094-44-01108-7.
- [KTR2005] Kharaghani, Hadi; Tayfeh-Rezaie, Behruz. A Hadamard matrix of order 428. J. Combinatorial Designs 13 (2005), no. 6, 435–440. DOI: 10.1002/jcd.20043.
- [DGK2014] Đoković, Dragomir Ž.; Golubitsky, Oleg; Kotsireas, Ilias S. Some new orders of Hadamard and skew-Hadamard matrices. J. Combinatorial Designs 22 (2014), no. 6, 270–277. DOI: 10.1002/jcd.21358. Preprint: arXiv:1301.3671
Contribution notes
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