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 〈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′∩〈x〉 or C _G (x)′∩〈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.