We suppose an orthogonal circulant autocorrelation. Then, the function is defined as a discrete circulant function $f(k)\in\mathbf{C}^{n}$ and $f(n+k)=f(k)$. The following convolution is indicated an arbitrary complex function and own conjugation:
$$ \sum_{k=0}^{n-1} \overline{f(k)}f(k+m)= \begin{cases} 1 & (m=0,n)\\ \\ 0 & (\text{otherwise}) \end{cases} $$
Then, the function can be expressed as $f(k)=r_{k}e^{i\theta_{k}}=z_{k}$. Therefore, the discrete convolution can be written by a matrix and vectors:
$$ \begin{bmatrix} \bar{z}_{0}z_{0} & \bar{z}_{1}z_{1} & \bar{z}_{2}z_{2} & \cdots & \bar{z}_{n-1}z_{n-1} \\ \bar{z_{0}}z_{1} & \bar{z_{1}}z_{2} & \bar{z_{2}}z_{3} & \cdots & \bar{z}_{n-1}z_{1} \\ \bar{z_{0}}z_{2} & \bar{z_{1}}z_{3} & \bar{z_{2}}z_{4} & \cdots & \bar{z}_{n-1}z_{2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \bar{z_{0}}z_{n-1} & \bar{z_{1}}z_{0} & \bar{z_{2}}z_{1} & \cdots & \bar{z}_{n-1}z_{n} \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} $$
Hence we can obtain a following phase degree system equation.
Main Question: $$ \begin{cases} \sum_{i=1}^n r_{i}r_{i+1}&\exp(i(\theta_{i}-\theta_{i+1}))&=0 \\ \sum_{i=1}^n r_{i}r_{i+2}&\exp(i(\theta_{i}-\theta_{i+2}))&=0 \\ \\ \ \ \ \ \vdots & \vdots &\vdots\\ \\ \sum_{i=1}^n r_{i}r_{i+n-1}&\exp(i(\theta_{i}-\theta_{i+n-1}))&=0 \\ \end{cases} $$
where $r_{i}$ are already known, $n$ kinds of unknown parameters $\theta_{i+n}=\theta_{i} \ (i=1,2,\cdots ,n)$, and the $n-1$ conditions are given like above.
My examination:
Essentially, $\lfloor n/2 \rfloor$ independent equations are given. Then, we can observe the above equation geometrically. For instance, when $n=5$, these equation can be described a diagram on the complex coordinate system and two closed shapes are obtained.
$$ \begin{cases} r_{1}r_{2}e^{i(\theta_{1}-\theta_{2})}+ r_{2}r_{3}e^{i(\theta_{2}-\theta_{3})}+ r_{3}r_{4}e^{i(\theta_{3}-\theta_{4})}+ r_{4}r_{5}e^{i(\theta_{4}-\theta_{5})}+ r_{5}r_{1}e^{i(\theta_{5}-\theta_{1})}=0 \\ r_{1}r_{3}e^{i(\theta_{1}-\theta_{3})}+ r_{2}r_{4}e^{i(\theta_{2}-\theta_{4})}+ r_{3}r_{5}e^{i(\theta_{3}-\theta_{5})}+ r_{4}r_{1}e^{i(\theta_{4}-\theta_{1})}+ r_{5}r_{2}e^{i(\theta_{5}-\theta_{2})}=0 \\ \end{cases} $$
Also, we can focus on only phase degrees. There are five nodes which have no concept of distance. For example, $r_{k}r_{j}e^{i(\theta_{k}-\theta_{j})}$ is defined a transfer operation between the starting point as node $j$ and the ending point as node $k$. When this operation is repeated over two times, this is same as linear combination of vectors. Based on these properties, we attempt to construct a graph and the following complete graph is obtained. Then, blue line and red line are expressed each roots and can be transfered the only single color. Further, each roots are established reversibly.
Apparently, it seems that there is a deep relation between circulant autocorrelation and topology.
