Difference sets with few character values

Abstract

The known families of difference sets can be subdivided into three classes: difference sets with Singer parameters, cyclotomic difference sets, and difference sets with gcd $$(v,n)>1$$ . It is remarkable that all the known difference sets with gcd $$(v,n)>1$$ have the so-called character divisibility property. Jungnickel and Schmidt (Difference sets: an update. London Math. Soc. Lecture Note Ser., vol. 245, pp. 89–112, Cambridge University Press, Cambridge 1997) posed the problem of constructing difference sets with gcd $$(v,n)>1$$ that do not satisfy this property. In an attempt to attack this problem, we use difference sets with three nontrivial character values as candidates, and get some necessary conditions.