Paley's theorem

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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.

[edit] 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. However, Hadamard matrices have been shown to exist for all $m \equiv 0 \mathrm{\,mod\,} 4$ for $m < 668$ .