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

Abstract

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.

10 pages, 2 tables