Hadamard Matrix

A Hadamard matrix is a square matrix whose entries are 1 and and whose rows are mutually orthogonal invented by Sylvester (1867) under the name of anallagmatic pavement, 26 years before Hadamard (1893) considered them. In a Hadamard matrix, placing any two columns or rows side by side gives half the adjacent cells the same sign and half the other sign. When viewed as pavements, cells with 1s are colored black and those with s are colored white. The first illustration shows orders 2, 4, 8, and 12 in the top row and 16, 20, 24, and 28 in the bottom row. Dense patterns can appear gray when reduced. The total numbers of black and white squares depend on the chosen matrix.

A Hadamard matrix of order is a solution to Hadamard's maximum determinant problem, i.e., has the maximum possible determinant (in absolute value) of any complex matrix with elements (Brenner and Cummings 1972), namely . An equivalent definition of the Hadamard matrices is given by

(1)

where is the identity matrix.

A Hadamard matrix of order corresponds to a Hadamard design (, , ), and a Hadamard matrix gives a graph on vertices known as a Hadamard graph.

The illustration above shows orders 2, 4, 8, and 16 in the top row and 32, 64, 128, and 256 in the bottom row. A complete set of Walsh functions of order gives a Hadamard matrix (Thompson et al. 1986, p. 204; Wolfram 2002, p. 1073). Hadamard matrices can be used to make error-correcting codes, in particular, the Reed-Muller code.

If and are known, then can be obtained by replacing all 1s in by and all s by . For , Hadamard matrices with , 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, and 100 cannot be obtained as nontrivial Kronecker products of lower order Hadamard matrices.

			(2)
			(3)
			(4)
			(5)

can be similarly generated from .

Hadamard (1893) remarked that a necessary condition for a Hadamard matrix to exist is that , 2, or a positive multiple of 4 (Brenner and Cummings 1972). Paley's theorem guarantees that there always exists a Hadamard matrix when is divisible by 4 and of the form for some positive integer , nonnegative integer , and an odd prime. In such cases, the matrices can be constructed using a Paley construction (Wolfram 2002, p. 1073). The number of Paley classes for , 8, ... are 2, 3, 2, 3, 2, 3, 2, 4, 1, ... (OEIS A074070). The values of for which there are no Paley classes (and hence cannot be constructed using a Paley construction) are 92, 116, 156, 172, 184, 188, 232, 236, 260, 268, ... (OEIS A046116).

It is conjectured that exists for all divisible by 4. Sawade (1985) constructed , illustrated above at left. Brouwer et al. (1989, p. 20) and van Lint and Wilson (1993) reported constructions for all divisible by 4 up to . Kharaghani and Tayfeh-Rezaie (2005) subsequently constructed , making 668 the smallest unresolved order at the time, as subsequently recorded by Cati and Pasechnik (2025). Epoch AI (2026) reported that L. Alpöge, P. Voinov, and S. Reynolds-Haertle had announced constructions obtained with Claude for the twelve previously unresolved orders below 2000: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964. The matrix is illustrated above at right. Thus constructions are now available for every positive order below 2000 divisible by 4. However, the proof of the general conjecture remains an important problem in coding theory.

The numbers of Hadamard matrices up to row and column permutations and sign changes, of orders for , 2, ... are 1, 1, 1, 5, 3, 60, 487, ... (OEIS A007299; Wolfram 2002, p. 1073). Djoković (2010) corrected the list in Colbourn and Dinitz (2007), including the order 764 constructed in 2008, and supplied constructions at the previously unresolved orders 23068, 28324, and 32996.

For real circulant matrices, OpenAI (2026) announced that a Hadamard matrix exists iff its order is 1 or 4. The manuscript also claims the resulting complete classification of Barker codes. This is distinct from the general Hadamard conjecture, which concerns matrices without the circulant restriction. The release proofs have not been independently verified here.

Egan et al. (2026) independently verified the twelve August constructions and constructed Hadamard matrices at sixteen orders that had been unresolved in the published literature. Several of those orders had also been settled independently. Their updated list leaves 95 unknown orders at most 10000, the smallest being 2396. Egan et al. (2026) report Claude assistance with construction searches, verification code, and mathematical arguments.