Nonexistence of Strong External Difference Families in Abelian Groups of Order Being Product of At Most Three Primes

TL;DR

This paper proves the nonexistence of strong external difference families in abelian groups of order being the product of up to three primes, with a specific exception involving large prime powers.

Contribution

It establishes a broad nonexistence result for strong external difference families in abelian groups of certain orders, narrowing the cases where such families can exist.

Findings

No strong external difference families exist in most abelian groups of order with up to three prime factors.

Exception potentially exists for groups of form C_p^3 with p > 3×10^{12}.

Significant progress in understanding the structure and limitations of external difference families.

Abstract

Let $v$ be a product of at most three not necessarily distinct primes. We prove that there exists no strong external difference family with more than two subsets in abelian group $G$ of order $v$, except possibly when $G=C_p^3$ and $p$ is a prime greater than $3 \times 10^{12}$.