On difference sets with small $\lambda$

TL;DR

This paper extends a multiplier-based method to efficiently prove the nonexistence of certain difference sets with small λ, significantly advancing computational verification of the Prime Power Conjecture and related parameters.

Contribution

It generalizes a 1989 multiplier argument to a broad range of parameters, enabling large-scale computational nonexistence proofs for difference sets.

Findings

Confirmed the Prime Power Conjecture up to order 2×10^{10}

Extended previous computations for λ=2 and k≤5000 up to 10^{10}

Proved nonexistence for many new parameter sets

Abstract

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 $(v,k,\lambda)$, 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 $k \leq 5000$ up to $10^{10}$, and show nonexistence for many other parameters.