On difference sets in high exponent 2-groups

We investigate the existence of difference sets in particular 2-groups. Being aware of the famous necessary conditions derived from Turyn's and Ma's theorems, we develop a new method to cover necessary conditions for the existence of (2(2d+2),2(2d+1)-2 (d) ,2(2d) -2 (d) ) difference sets, for some large classes of 2-groups. If a 2-group G possesses a normal cyclic subgroup aOE (c) x > of order greater than 2 (d+3+p) , where the outer elements act on the cyclic subgroup similarly as in the dihedral, semidihedral, quaternion or modular groups and 2 (p) describes the size of G'a (c) aOE (c) x > or C (G) (x)'a (c) aOE (c) x >, then there is no difference set in such a group. Technically, we use a simple fact on how sums of 2 (n) -roots of unity can be annulated and use it to characterize properties of norm invariance (prescribed norm). This approach gives necessary conditions when a linear combination of 2 (n) -roots of unity remains unchanged under homomorphism actions in the sense of the norm.