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 $$\vert b\cdot b'\vert = \frac{1}{\sqrt d}$$ ($b\cdot b'$ 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.
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机译：在我们之前的论文\ cite {co1}中，我们证明了循环矩阵的理论可以恢复以下结果：存在维数为$ p $的$ p + 1 $互不偏基，$ p $是任意质数。如果$ \ forall b \ in \ mathcal B'\\ forall b'\ in \ mathcal B'$个中的$ \ forall b \，则说两个正交基数$ \ mathcal B'\\\ mathbb C ^ d $的B'$相互无偏具有$$ \ vert b \ cdot b'\ vert = \ frac {1} {\ sqrt d} $$（$ \ mathbb C ^ d $中的$ b \ cdot b'$埃尔米特量积）。在本文中，我们证明了具有循环块的块循环矩阵理论允许非常简单地显示以下已知结果：如果$ d = p ^ n $（$ p $为质数，$ n $为任意整数），则存在$ d + 1 $个在$ \ mathbb C ^ d $中的互不偏基。我们的结果在很大程度上取决于Klimov，Munoz，Romero \ cite {klimuro}的想法。作为子产品，我们以$ p \ ge 3 $的形式恢复二次Weil和的性质，这概括了以下事实：在最佳情况下，二次高斯和性质取决于我们的结果。
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