Negative Latin-Square-Type Partial Difference Sets in Non-Elementary Abelian 2-Groups

Using cubic cyclotomic classes, character theory, and product constructions motivated by generalized Denniston partial difference sets, we construct negative Latin-square-type partial difference sets in $\mathbb{F}_{2^6}^{+}\times\mathbb{Z}_4^4$, $\mathbb{Z}_4^4\times\mathbb{F}_{2^{10}}^{+}$, $\mathbb{F}_{2^{10}}^{+}\times\mathbb{Z}_8^4$, $\mathbb{Z}_8^4\times\mathbb{F}_{2^{14}}^{+}$, and $\mathbb{Z}_4^4\times\mathbb{Z}_{16}^2$.

The first four constructions replace cubic cyclotomic partitions by partitions of non-elementary abelian 2-groups having the same character-value patterns.

To the best of our knowledge, these are the first partial difference sets with the stated parameters in the indicated groups.