Constructions of Nested Partial Difference Sets with Galois Rings

Abstract

There have been several recent constructions of partial difference sets (PDSs) using the Galois rings GR(p^2, t) for p a prime and t any positive integer. This paper presents constructions of partial difference sets in (Z_{p^r})^{2t} where p is any prime, and r and t are any positive integers. For the case where r > 2 many of the partial difference sets are constructed in groups with parameters distinct from other known constructions, and the PDSs are nested. Another construction of Paley partial difference sets is given for the case when p is odd. The constructions make use of character theory and of the structure of the Galois ring GR(p^r, t), and in particular, the ring GR(p^r, t) \times GR(p^r, t). The paper concludes with some open related problems.