Partial difference sets from unions of cyclotomic classes

In their study of two-weight irreducible cyclic codes, Schmidt and White (2002) obtained a necessary and sufficient condition on $(q,N)$ under which the multiplicative subgroup of index N of the finite field $F_q$ forms a regular partial difference set (PDS) in the additive group of $F_q$. They also found 11 sporadic examples by a computer search aside from two known infinite families of PDS. In this paper, we study the problem of determining for which $(q,N)$ a union of multiple cosets of the multiplicative subgroup of index N of $F_q$ forms a regular PDS in the additive group of $F_q$. Building on the work of Schmidt and White, we find a necessary and sufficient numerical condition on the parameters $(q,N)$ for unions of multiple cyclotomic classes to form regular PDS in $(F_q,+)$. We then apply the theorem to the situation where unions of a small number of classes are selected in a structured manner. We obtain a new infinite family of regular PDS not belonging to previously known families, and two sporadic examples of regular PDS (one of which is new) with the help of a computer research. We further propose a conjecture analogous to the Schmidt-White conjecture proposed in their 2002 paper.