On an Additive Characterization of a Skew Hadamard (n, n−1/ 2 , n−3 4 )-Difference Set in an Abelian Group

We give a combinatorial proof of an additive characterization of a skew Hadamard (n, n−1 2 , n−3 4 )-difference set in an abelian group G. This research was motivated by the p = 4k + 3 case of Theorem 2.2 of Monico and Elia [4] concerning an additive characterization of quadratic residues in Z p. We then use the known classification of skew (n, n−1 2 , n−3 4 )-difference sets in Z n to give a result for integers n = 4k +3 that strengthens and provides an alternative proof of the p = 4k + 3 case of Theorem 2.2 of [4]