Negative Latin square type partial difference sets in nonelementary abelian 2-groups

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groups Z2k4 × Z4−4k2 for all k when is odd and for all k < when is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all k when is even and for all k < when is odd. These constructions provide the first example where the non-homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new. 1