Negative Latin Square type Partial Difference Sets in Nonelementary Abelian 2-Groups

Combining results on quadrics in projective geometries withan algebraic interplay between finite fields and Galois rings,the first known family of partial difference sets with negativeLatin square type parameters is constructed in nonelementaryabelian groups, the groups x for all k when is odd andfor all k < when is even. Similarly, partial differencesets with Latin square type parameters are constructed in thesame groups for all k when is even and for all k< when is odd. These constructions provide the first example wherethe non-homomorphic bijection approach outlined by Hagita andSchmidt can produce difference sets in groups that previouslyhad no known constructions. Computer computations indicate thatthe strongly regular graphs associated to the partial differencesets are not isomorphic to the known graphs, and it is conjecturedthat the family of strongly regular graphs will be new.