New hadamard matrices of order 4p^2 obtained from Jacobi sums of order 16

Abstract

Let p=7 mod 6 be a prime. Then there are integers a,b,c,d with a=15 mod 6, b= 0 mod 4, p^2=a^2+2(b^2+c^2+d^2), and 2ab=c^2-2cd-d^2. We show that there is a regular Hadamard matrix of order 4p2 provided that p=a±2b or p=a+δ12b+4δ2c+4δ1δ2d with δi=±1.