A Hadamard matrix is a type of square(-1,1)-matrix 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. Therefore, the Hadamard matrix must have white squares (s) and black squares (1s).
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
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 . As of 1993, Hadamard matrices were known for all divisible by 4 up to (Brouwer et al. 1989, p. 20; van Lint and
Wilson 1993). A
was subsequently constructed (Kharaghani and Tayfeh-Rezaie 2004), illustrated above,
leaving the smallest unknown order as 668. However, the proof of the general conjecture
remains an important problem in coding theory. 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
divisible by 4 for which it is possible to construct a Hadamard matrix: 764, 23068,
28324, 32996.
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