On abelian -difference sets

In this paper we prove that an abelian group contains ( 2 2 m + 1 ( 2 m − 1 + 1 ) , 2 m ( 2 m + 1 ) , 2 m ) -difference sets with m ⩾ 3 if and only if it contains an elementary abelian 2-group of order 2 2 m . Our proof shows that the method of constructing such difference sets is essentially unique.