Non‐existence of (pm,k,1)$(p^m, k, 1)$ difference sets

Abstract

Let G $G$ be an additive abelian group of size v $v$ . A k $k$ -subset D $D$ of G $G$ is called a ( v , k , λ ) $(v, k, \lambda )$ -difference set if every non-identity element in G $G$ can be written in λ $\lambda$ ways as the difference of two elements in D $D$ . This letter proves the non-existence of ( p m , k , 1 ) $(p^m, k, 1)$ -difference sets, for all prime p $p$ and m > 1 $m>1$ .