On the Existence of Abelian Hadamard Difference Sets and a New Family of Difference Sets

We present a construction of Hadamard difference sets in abelian groups of order 4p4n, whose Sylowp-subgroups are elementary. By a standard composition procedure, we can now conclude that (4h2, 2h2−h,h2−h)-Hadamard difference sets exist forh= 2ε13ε2u2, where ε1, ε2= 0 or 1 anduis a positive integer. We then generalize the construction of Hadamard difference sets to construct a family of (4q2n(q2n− 1)/(q2−1),q2n−1[2(q2n− 1)/(q+ 1) + 1], (q2n−q2n−1)(q2n−1 + 1)/(q+ 1)-difference sets, whereqis an even power of an odd prime or any power of 3.