Constructions of Hadamard Difference Sets

Using a spread ofPG(3, p) and certain projective two-weight codes, we give a general construction of Hadamard difference sets in groupsH×(Zp)4, whereHis either the Klein 4-group or the cyclic group of order 4, andpis an odd prime. In the casep≡3 (mod 4), we use an ovoidal fibration ofPG(3, p) to construct Hadamard difference sets, this construction includes Xiaʹs construction of Hadamard difference sets as a special case. In the casep≡1 (mod 4), we construct new reversible Hadamard difference sets by explicitly constructing the two-weight codes needed in our general construction method. Using a well-known composition theorem, we conclude that there exist Hadamard difference sets with parameters (4m2, 2m2−m, m2−m), wherem=2a3b52c1132c2172c3p21p22…p2twitha, b, c1, c2, c3positive integers and where eachpjis a prime congruent to 3 modulo 4, 1⩽j⩽t.