On difference sets with small $$\lambda $$

Abstract

In a 1989 paper, Arasu (Arch Math 53:622–624, 1989) used an observation about multipliers to show that no (352, 27, 2) difference set exists in any abelian group. The proof is quite short and required no computer assistance. We show that it may be applied to a wide range of parameters \((v,k,\lambda )\), particularly for small values of \(\lambda \). With it, a computer search was able to show that the Prime Power Conjecture is true up to order \(2 \cdot 10^{10}\), extend Hughes and Dickey’s computations for \(\lambda =2\) and \(k \le 5000\) up to \(10^{10}\), and show nonexistence for many other parameters.