Schmidt: New Hadamard Matrices of Order 4p 2 obtained from Jacobi Sums of Order 16

Let p ≡ 7 mod 16 be a prime. Then there are integers a, b, c, d with a ≡ 15 mod 16, b ≡ 0 mod 4, p2 = a2 + 2(b2 + c2 + d2), and 2ab = c2 − 2cd − d2. We show that there is a regular Hadamard matrix of order 4p2 provided that p = a ± 2b or p = a+ δ1b+4δ2c+4δ1δ2d with δi = ±1. ∗This research was done during a visit of the first two authors at the University of Augsburg 1