Difference sets in abelian 2-groups

If G is an abelian group of order u, and D is a subset of G with k elements such that every nonidentity element can be expressed 2 times in the form a b, where a and b are elements of D, then D is called a (u, k, A) difference set in G. The order n of the difference set is k 1. In this paper we consider the parameter values v = 22di 2, k = 22d+ ’ 24 ,I= 22d 2d, and n=22d. The rank r of G is the minimum number of generators, and the exponent (exp(G) = 2’) is the size of the largest cyclic subgroup of G. For a given order 22d+2 of G, (2d+2)/e<r<2d+3-e. In terms of r and e, the current state of knowledge can be summarized as follows: (1) If e 2 d + 3, then G does not have a difference set (Turyn [S]); (2) if rad+ 1, then G does have a difference set (Dillon [2]). Graphically, see Fig. 1. The following is a result of Turyn [S], and it will be the main tool of this paper: