New Hadamard matrices of order 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 , p 2 = a 2 + 2 ( b 2 + c 2 + d 2 ) , and 2 ab = c 2 - 2 cd - d 2 . We show that there is a regular Hadamard matrix of order 4 p 2 provided that p = a ± 2 b or p = a + δ 1 2 b + 4 δ 2 c + 4 δ 1 δ 2 d with δ i = ± 1 .