Paley partial difference sets in groups of order n4 and 9n4 for any odd n>1

A partial difference set with parameters 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 n4 and 9n4. We are also able to construct Paley–Hadamard difference sets of the Stanton–Sprott family in groups of order n4(n4±2) when n4±2 is a prime power and 9n4(9n4±2) when 9n4±2 is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices.