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In our previous paper \cite{co1} we have shown that the theory of circulantmatrices allows to recover the result that there exists $p+1$ Mutually UnbiasedBases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormalbases $\mathcal B, \mathcal B'$ of $\mathbb C^d$ are said mutually unbiased if$\forall b\in \mathcal B, \forall b' \in \mathcal B'$ one has that $$| b\cdotb'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ hermitian scalar product in $\mathbbC^d$). In this paper we show that the theory of block-circulant matrices withcirculant 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 UnbiasedBases 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 Weilsums for $p\ge 3$, which generalizes the fact that in the prime case thequadratic Gauss sums properties follow from our results.

Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case