The Hadamard conjecture: existence of Hadamard matrices of every order 4m

The Hadamard conjecture asserts that for every positive integer m there exists a Hadamard matrix of order 4m, i.e. a 4m by 4m matrix with entries plus or minus 1 whose rows are mutually orthogonal, so that H times H-transpose equals 4m times the identity. Open since 1893, it is the central existence problem of combinatorial design theory and determines when Hadamard's determinant bound is attained exactly. A 2021 paper on binary Hadamard codes and their s-PD sets cites the conjecture with 668 as the smallest then-unresolved order. The conjecture itself remains open: an August 2026 announcement produced explicit matrices for all previously unknown orders up to 2000, but no construction or nonexistence argument is known for general m.

Background

A Hadamard matrix is a square $n \times n$ matrix $H$ with entries in $\{\pm 1\}$ whose distinct rows are orthogonal, equivalently $HH^T = nI$ where $I$ is the identity. Such matrices arise from Hadamard's 1893 theorem on determinants: if every entry of a real matrix has absolute value at most 1, then $|\det M| \le n^{n/2}$, with equality exactly for Hadamard matrices (up to scaling). Comparing three rows column by column shows that for $n > 2$ the order must be divisible by 4, so orders 1, 2 and the multiples of 4 are the only possibilities. The Hadamard conjecture states that this divisibility condition is the only obstruction: a Hadamard matrix of order $4m$ should exist for every positive integer $m$.

Two classical infinite families are known: Sylvester (1867) observed that replacing each entry of a Hadamard matrix by its $2 \times 2$ block form doubles the order, giving matrices of every order $2^k$; Paley (1933) used quadratic residues and non-residues in finite fields of odd prime-power order $q$ to build Hadamard matrices of orders $q+1$ and $2(q+1)$. Taking tensor products of known matrices yields further composite orders, but these families still leave most multiples of 4 uncovered. The remaining progress has come from design-theoretic array constructions and computer search: for example, Goethals-Seidel arrays assemble a Hadamard matrix of order $4n$ from four $n \times n$ matrices $A, B, C, D$, each circulant (every row a cyclic shift of the one above) or symmetric circulant, satisfying $AA^T + BB^T + CC^T + DD^T = 4nI$, and Williamson-type searches hunt for such quadruples. These methods resolve individual orders but give no uniform construction.

The history of small orders tracks the conjecture's frontier. After Kharaghani and Tayfeh-Rezaie constructed a Hadamard matrix of order 428 in 2005, the order 668 became the smallest without any known example, and it held that status for two decades: Djokovic resolved 764 in 2007 and, with Golubitsky and Kotsireas, 1004 and 2524 in 2013; Balonin and Djokovic added the first symmetric matrices of orders 268, 412, 436 and 604 in 2018; searches of Williamson type continue in 2026 (Kharaghani, Mohammadian and Tayfeh-Rezaie completed the census of near-Williamson quadruples for all odd orders up to 35 and showed existence up to 63). In August 2026 an announcement by Alpoge, Voinov, Reynolds-Haertle and Claude (recorded in the On-Line Encyclopedia of Integer Sequences (OEIS) entry A007299) produced explicit matrices for all twelve multiples of 4 up to 2000 that were previously unresolved, including 668.

Coding-theoretically, Hadamard matrices carry error-correcting codes: the rows of a Hadamard matrix, mapped from $\{\pm 1\}$ to bits, form the codewords of a Hadamard code, a binary error-correcting code whose minimum distance is half its length. Chathely and Deore (2021) construct binary linear Hadamard codes with parameters $(2n, 4n, n)$, generalized over the finite field $F_{2^k}$ (the field with $2^k$ elements) to parameters $(2^{k-1}n, 2^k n, 2^{k-2}n)$, and build s-PD sets for them — subsets of the permutation automorphism group of a code of size sufficient to move any error pattern of weight at most s out of the information positions, which enables permutation decoding of s errors; they achieve the minimal possible size s+1 for their codes over $F_4$ (the field with four elements). Every such construction presupposes a Hadamard matrix of the underlying order, so the Hadamard conjecture is exactly the existence question that would make Hadamard codes, and the designs and maximal determinants derived from Hadamard matrices, available at every admissible order.

Problem Statement

For every positive integer m, does there exist a Hadamard matrix of order 4m — a $4m \times 4m$ matrix $H$ with entries in $\{\pm 1\}$ satisfying $HH^T = 4m I$? An affirmative answer must establish existence for all m at once, either by a uniform explicit construction (a formula or algorithm) proved valid for every m, or by any other complete proof of universal existence. A negative answer must prove that some specific order $4m$ admits no Hadamard matrix at all. Explicit matrices for individual orders settle instances but not the question: announced examples exist for every multiple of 4 up to 2000, yet infinitely many orders remain without any construction and the universal question is untouched. The problem may equally be phrased through Hadamard's determinant bound: is the maximum $|\det M|$ over all $n \times n$ matrices with entries of absolute value at most 1 equal to $n^{n/2}$ for every order $n$ divisible by 4?

References

1. J. Hadamard, 'Résolution d'une question relative aux déterminants', Bulletin des Sciences Mathématiques (2) 17 (1893) 240-246. zbMATH JFM 25.0221.02, https://zbmath.org/2680844
2. J. J. Sylvester, 'Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers', The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 34 (1867). DOI 10.1080/14786446708639914, https://doi.org/10.1080/14786446708639914
3. R. E. A. C. Paley, 'On Orthogonal Matrices', Journal of Mathematics and Physics 12 (1933) 311-320. DOI 10.1002/sapm1933121311, https://doi.org/10.1002/sapm1933121311
4. H. Kharaghani and B. Tayfeh-Rezaie, 'A Hadamard matrix of order 428', Journal of Combinatorial Designs 13 (2005) 435-440. DOI 10.1002/jcd.20043, https://doi.org/10.1002/jcd.20043
5. D. Z. Djokovic, 'Hadamard matrices of order 764 exist', arXiv:math/0703312 [math.CO] (2007), https://arxiv.org/abs/math/0703312
6. D. Z. Djokovic, O. Golubitsky and I. S. Kotsireas, 'Some new orders of Hadamard and skew-Hadamard matrices', arXiv:1301.3671 [math.CO] (2013), https://arxiv.org/abs/1301.3671
7. N. A. Balonin and D. Z. Djokovic, 'Symmetric Hadamard matrices of orders 268, 412, 436 and 604', arXiv:1803.08787 [math.CO] (2018), https://arxiv.org/abs/1803.08787
8. B. J. Chathely and R. P. Deore, 'Construction of binary Hadamard codes and their s-PD sets', Cryptography and Communications 13(3) (2021) 425-438. DOI 10.1007/s12095-021-00471-5, https://doi.org/10.1007/s12095-021-00471-5
9. H. Kharaghani, A. Mohammadian and B. Tayfeh-Rezaie, 'A search for Hadamard matrices of Williamson type', arXiv:2605.08661 [math.CO] (2026), https://arxiv.org/abs/2605.08661
10. A. F. Ramos, D. B. Hulak and R. J. G. B. de Queiroz, 'Multiplier obstructions for Legendre pairs of length 333', arXiv:2607.20765 [math.CO] (2026), https://arxiv.org/abs/2607.20765
11. OEIS Foundation Inc., 'The On-Line Encyclopedia of Integer Sequences, entry A007299: number of Hadamard matrices of order 4n' (with Hadamard-conjecture status update of 12 August 2026), https://oeis.org/A007299
12. L. Alpoge, P. Voinov, S. Reynolds-Haertle and Claude, 'Hadamard matrix of order 668' (announcement, August 2026; explicit 668 x 668 +/- matrix artifact), pastebin ID mCFde59D, https://pastebin.com/mCFde59D