A generalization of circulant Hadamard and conference matrices

We study the existence and construction of circulant matrices C of order n >= 2 with diagonal entries d >= 0, off-diagonal entries +/- 1 and mutually orthogonal rows. These matrices generalize circulant conference (d = 0) and circulant Hadamard (d = 1) matrices. We demonstrate that matrices C exist for every order n and for d chosen such that n = 2d + 2, and we find all solutions C with this property. Furthermore, we prove that if C is symmetric, or n-1 is prime, or d is not an odd integer, then necessarily n = 2d+2. Finally, we conjecture that the relation n = 2d + 2 holds for every matrix C, which generalizes the circulant Hadamard conjecture. We support the proposed conjecture by computing all the existing solutions up to n = 50. (C) 2019 Elsevier Inc. All rights reserved.