Paley's theorem (Paley 1933) guarantees the existence of certain Hadamard matrices. If 
is an odd prime or 
and 
is any positive integer,
then there is a Hadamard matrix of order
 |
where 
is any positive integer such that 
. If 
is of this form, the matrix can
be constructed with a Paley construction. If
is divisible by 4 but not of
the form above, the Paley class is undefined.
However, Epoch AI (2026) reported constructions establishing existence for every
positive 
satisfying 
.
