  Mutually unbiased bases that can be cyclically generated by a single unitary
operator are of special interest, since they can be readily implemented in
practice. We show that, for a system of qubits, finding such a generator can be
cast as the problem of finding a symmetric matrix over the field $\mathbb{F}_2$
equipped with an irreducible characteristic polynomial of a given Fibonacci
index. The entanglement structure of the resulting complete sets is determined
by two additive matrices of the same size.
