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OP-12029OpenCombinatorics

Smallest unsolved instance of the Hadamard conjecture (C23aC_{23a})

Smallest unsolved instance of the Hadamard conjecture: determine C23aC_{23a} or improve its bounds; the repository records 668 (lower) and ∞\infty (upper).

MSC 05B20 · Combinatorial aspects of matrices (incidence, Hadamard, etc.)

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

A Hadamard matrix of order nn is an n×nn\times n matrix HH with entries in {−1,1}\{-1,1\} such that

HH⊤=nIn.HH^{\top} = n I_n.

Equivalently, the rows (and columns) are pairwise orthogonal.

It is known that Hadamard matrices can exist only for n=1,2n=1,2, or n≡0(mod4)n\equiv 0 \pmod{4}.

We define C23aC_{23a} to be the smallest integer n≡0(mod4)n\equiv 0 \pmod{4} such that there is no Hadamard matrix of order nn. If no such nn exists, we set C23a=∞C_{23a}=\infty.

The Hadamard conjecture asserts that C23a=∞C_{23a}=\infty, i.e. that Hadamard matrices exist for every order n≡0(mod4)n\equiv 0 \pmod{4}.

(Equivalently, by Hadamard’s determinant inequality, for A∈{−1,1}n×nA\in\{-1,1\}^{n\times n} one has ∣det⁡(A)∣≤nn/2|\det(A)|\le n^{n/2}, with equality iff AA is Hadamard; the conjecture predicts equality is attainable for all n≡0(mod4)n\equiv 0\pmod4.)

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Background

  • A benchmark open instance is the existence of a Hadamard matrix of order 668668; in the range n≤1208n\le 1208, the only unresolved orders are 668,716,892,1132668,716,892,1132 [CP2024].
  • Classical infinite families of Hadamard orders include:
  • Sylvester’s Kronecker-product construction, giving Hadamard matrices of order 2m2^m for every m≥0m\ge 0 [Syl1867].
  • Paley’s constructions, giving Hadamard matrices of order q+1q+1 when qq is a prime power with q≡3(mod4)q\equiv 3\pmod{4}, and of order 2(q+1)2(q+1) when qq is a prime power with q≡1(mod4)q\equiv 1\pmod{4} [Pal1933].
  • Wikipedia page on Hadamard matrices
  • SageMath implementation and access to Sloane’s library: SageMath documentation

References (from the repository page)

  • [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

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 668; best known upper bound ∞\infty. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
∞\infty Trivial No finite upper bound is known; conjecturally sharp (Hadamard conjecture).

Known lower bounds

Bound Reference Comments
44 Trivial By definition C23aC_{23a} (if finite) is a multiple of 44.
668668 [CP2024] All orders n<668n<668 with n≡0(mod4)n\equiv 0\pmod4 are known to admit Hadamard matrices; the smallest currently unresolved order is 668668.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Smallest unsolved instance of the Hadamard conjecture ($C_{23a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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