On difference sets with small $λ$

In a 1989 paper \cite{arasu2}, Arasu 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 $(v,k,λ)$, particularly for small values of $λ$. 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 $λ=2$ and $k \leq 5000$ up to $10^{10}$, and show nonexistence for many other parameters.

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