In this paper, we prove that a binary sequence is perfect (resp., quasi-perfect)if and only if its support set for any finite group (not necessarily cyclic) is a Hadamard difference set of type Ⅰ (resp., type Ⅱ); and we prove that the keel of any nonzero linear functional (or the image of any linear transformation A with dim(Ker A) = 1) on the linear space GF(2m) over the field GF(2) (excluding 0) is a cyclic Hadamard difference set of type Ⅱ using Gaussian sums; and we prove that the multiplier group of the above difference set is equal to the Galois group Gal(GF(2m)/GF(2)); and we mention the relationship between the Hadamard transform and the irreducible complex characters.