Non-existence and Construction of Pre-difference Sets, and Equi-Distributed Subsets in Association Schemes

In the previous work, we introduce a notion of pre-difference sets in a finite group G defined by weaker conditions than the difference sets. In this paper we gave a construction of a pre-difference set in $$G=NA$$ with A an abelian subgroup and N a subgroup satisfying $$N\cap A=\{e\}$$ , from a difference set in $$N\times A$$ . This gives a (16, 6, 2) pre-difference set in $$D_{16}$$ and a (27, 13, 6) pre-difference set in UT(3, 3), where no non-trivial difference sets exist. We also give a product construction of pre-difference sets similar to Kesava Menon construction, which provides infinite series of pre-difference sets that are not difference sets. We show some necessary conditions for the existence of a pre-difference set in a group with index 2 subgroup. For the proofs, we use a rather simple framework “relation partitions,” which is obtained by dropping an axiom from association schemes. Most results are proved in that frame work.