Small circulant complex Hadamard matrices of Butson type

We study the circulant complex Hadamard matrices of order n whose entries are ith roots of unity. For n = l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n = p + q, l = pq with p, q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n, l. (C) 2015 Published by Elsevier Ltd.