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

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 $\mathcal{B},{\mathcal{B}}^{\mathrm{\prime }}$ of ${\mathbb{C}}^d$ are said mutually unbiased if $\mathrm{\forall }b\in \mathcal{B},\mathrm{\forall }{b}^{\mathrm{\prime }}\in {\mathcal{B}}^{\mathrm{\prime }}$ one has that $$\mathrm{\mid }b\cdot b^{\mathrm{\prime }}\mathrm{\mid }=\frac{1}{\sqrt{d}}$$ ($b\cdot b^{\mathrm{\prime }}$ 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.