Non-existence of partial difference sets in Abelian groups of order $$8p^3$$ 8 p 3

Abstract

In this paper we prove non-existence of nontrivial partial difference sets in Abelian groups of order $$8p^3$$ , where $$p\ge 3$$ is a prime number. These groups seemed to have the potential of admitting at least two infinite families of PDSs, and even the smallest case, $$p=3$$ , had been open for twenty years until settled recently by the authors and E. Neubert. Here, using the integrality and divisibility conditions for PDSs, we first describe all hypothetical parameter sets of nontrivial partial difference sets in these groups. Then we prove the non-existence of a PDS for each of these hypothetical parameter sets by combining a recent local multiplier result with some geometry and elementary number theory.