Williamson conjecture

In combinatorial mathematics, specifically in combinatorial design theory and combinatorial matrix theory, the Williamson conjecture is that Williamson matrices of order ${\displaystyle n}$ exist for all positive integers ${\displaystyle n}$. Four symmetric and circulant matrices ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$ are called Williamson matrices if their entries are ${\displaystyle \pm 1}$ and they satisfy the relationship

${\displaystyle A^{2}+B^{2}+C^{2}+D^{2}=4n\,I}$

where ${\displaystyle I}$ is the identity matrix of order ${\displaystyle n}$. John Williamson showed that if ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$ are Williamson matrices then

${\displaystyle {\begin{bmatrix}A&B&C&D\\-B&A&-D&C\\-C&D&A&-B\\-D&-C&B&A\end{bmatrix}}}$

is an Hadamard matrix of order ${\displaystyle 4n}$.¹ It was once considered likely that Williamson matrices exist for all orders ${\displaystyle n}$ and that the structure of Williamson matrices could provide a route to proving the Hadamard conjecture that Hadamard matrices exist for all orders ${\displaystyle 4n}$.² However, in 1993 the Williamson conjecture was shown to be false by Dragomir Ž. Ðoković through an exhaustive computer search, which demonstrated that Williamson matrices do not exist of order ${\displaystyle n=35}$.³ In 2008, the counterexamples 47, 53, and 59 were additionally discovered.⁴

Following the negative result of Ðoković, which ruled out the existence of Williamson matrices of order ${\displaystyle n=35}$, it was shown in 2019 that relaxing the symmetry and circulant requirements nevertheless permits a Hadamard matrix of this block form to exist for that order.⁵ One such instance is given by the sequences

a = --+--+--+++-+----+-+++--+--+-------
b = +---+---+-++-+--+-++-+---+---++++++
c = -++---+-+--+--++--+++++-+-+++++-+--
d = +----+++-+-+--++--+-+-+++----++---+

with the associated matrices defined by

A = circulant(a)
B = circulant(b)
C = fliplr(circulant(c))
D = circulant(d)