Paley construction

In mathematics, Paley's theorem is a theorem on Hadamard matrices. It was proved in 1933 and is named after the English mathematician Raymond Paley.

Statement of the theorem

Let $q$ be an odd prime or $0$. Let $n$ be a natural number. Then there exists a Hadamard matrix $H$ of order

$m=2^{k}(q^{n}+1),$

where $k$ is a natural number such that

$m\equiv 0\mathrm {\,mod\,} 4.$

If $m$ is of the above form, then $H$ can be constructed using a Paley construction. If $m$ is divisible by 4 but is not of the above form, then the Paley class is undefined. Currently, Hadamard matrices have been shown to exist for all $m\equiv 0\mathrm {\,mod\,} 4$ for $m<668$.

References

• Paley, R.E.A.C. (1933). "On orthogonal matrices". J. Math. Phys. 12: 311–320.