  In our previous paper \cite{co1} we have shown that the theory of circulant
matrices allows to recover the result that there exists $p+1$ Mutually Unbiased
Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal
bases $\mathcal B, \mathcal B'$ of $\mathbb C^d$ are said mutually unbiased if
$\forall b\in \mathcal B, \forall b' \in \mathcal B'$ one has that $$| b\cdot
b'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ hermitian scalar product in $\mathbb
C^d$). In this paper we show that the theory of block-circulant matrices with
circulant blocks allows to show very simply the known result that if $d=p^n$
($p$ a prime number, $n$ any integer) there exists $d+1$ mutually Unbiased
Bases in $\mathbb C^d$. Our result relies heavily on an idea of Klimov, Munoz,
Romero \cite{klimuro}. As a subproduct we recover properties of quadratic Weil
sums for $p\ge 3$, which generalizes the fact that in the prime case the
quadratic Gauss sums properties follow from our results.
