On abelian $(2^{2m+1}(2^{m-1}+1), 2^m(2^m+1), 2^m)$-difference sets

Abstract

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