Rings and constructions of partial difference sets (Q1406553)

The purpose of the paper is to construct (and better understand) PDSs on nonelementary abelian groups (and even with nonelementary Sylow subgroups). Let \(R\) be a finite Frobenius local ring with maximal ideal \(M\), and let \(I\) be a proper ideal of \(R\). Under suitable conditions a reversible PDS on (the additive group of) \(M/I\times M/I\) can be extended to a PDS on \(R/I\times R/I\). A second construction shows that in several cases a reversible PDS on \(M/r(M)\times R/r(M)\) (where \(r(M)\) is the right annihilator of \(M\) in \(R\)) can be extended to \(R\times R\). The third construction uses a chain ring, i.e. a ring \(R\) whose left ideals form a chain. Proofs are complex but very interesting and evocative; suggestions for natural questions and enlightening examples are given.