  Combining results on quadrics in projective geometries with an algebraic
interplay between finite fields and Galois rings, we construct the first known
family of partial difference sets with negative Latin square type parameters in
nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for
all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even.
Similarly, we construct partial difference sets with Latin square type
parameters in the same groups for all $k$ when $\ell$ is even and for all
$k<\ell$ when $\ell$ is odd. These constructions provide the first example that
the non-homomorphic bijection approach outlined by Hagita and Schmidt
\cite{hagitaschmidt} can produce difference sets in groups that previously had
no known constructions. Computer computations indicate that the strongly
regular graphs associated to the PDSs are not isomorphic to the known graphs,
and we conjecture that the family of strongly regular graphs will be new.
