Circulant 16-Modular Hadamard Matrices and Jacobi Sums

We are concerned here with the existence problem of 16-modular circulant Hadamard matrices H of size 4p (p prime), satisfying the additional condition that any two rows at distance n/2 in H are strictly orthogonal. A necessary existence condition is p≡1 mod 8. For p≡1 mod 16, existence follows from the more general result of a previous paper of ours, showing the existence of (p−1)-modular matrices of the above kind. In the remaining case p≡9 mod 16, we construct explicit examples which solve the problem whenever 2 is a fourth power mod p. When 2 is not a fourth power mod p, we conjecture that such matrices cannot exist.