On abelian (22m+1(2m−1+1),m2(m2+1),m2)-difference sets

In this paper we prove that an abelian group contains (22m+1(2m−1+1),m2(m2+1),m2)-difference sets with m⩾3 if and only if it contains an elementary abelian 2-group of order 22m. Our proof shows that the method of constructing such difference sets is essentially unique.