In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley.
The Paley construction uses quadratic residues in a finite field GF(q) where q is a power of an odd prime number. There are two versions of the construction depending on whether q is congruent to 1 or 3 modulo 4.