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 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.

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 error-correcting 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 built up from 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, as illustrated above (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 . 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 , illustrated above, leaving 668 as the smallest unknown order. However, the proof of the general conjecture remains an important problem in coding theory.

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. Thus constructions are now available for every positive order below 2000 divisible by 4. This finite set of constructions does not resolve the general conjecture.

The number of distinct Hadamard matrices of orders for , 2, ... are 1, 1, 1, 5, 3, 60, 487, ... (OEIS A007299; Wolfram 2002, p. 1073). Djoković (2009) corrected the list in Colbourn and Dinitz (2007) and found four previously unknown orders divisible by 4 for which it is possible to construct a Hadamard matrix: 764, 23068, 28324, 32996.