Paley partial difference sets in groups of order and for any odd

A partial difference set with parameters ( v , v − 1 2 , v − 5 4 , v − 1 4 ) is said to be of Paley type. In this paper, we give a recursive theorem that for all odd n > 1 constructs Paley partial difference sets in certain groups of order n 4 and 9 n 4 . We are also able to construct Paley–Hadamard difference sets of the Stanton–Sprott family in groups of order n 4 ( n 4 ± 2 ) when n 4 ± 2 is a prime power and 9 n 4 ( 9 n 4 ± 2 ) when 9 n 4 ± 2 is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices.