Consider a symmetric circulant matrix with entries in each diagonal , sub and superdiagonal being either $0$ or $1$. Example: $$\begin{bmatrix}1&1&0&1&0&1\\1&1&1&0&1&0\\0&1&1&1&0&1\\1&0&1&1&1&0\\0&1&0&1&1&1\\1&0&1&0&1&1\end{bmatrix}$$.
Now, I wish to replace the diagonal entries with a string of $n$ numbers, where $n$ be the order of the matrix; which consists of $k+1$ distinct numbers repeated in a certain order where $k$ be the number of nonzero super(sub) diagonals of matrix and the entries in the nonzero super(sub) diagonals with the same string of numbers permuted cyclically so that each row or column has exactly one instance of any nonzero number, and the order of the entries in the principal diagonal remains. I think this should be always possible because there $k$ rotations (cycles) of the entries in the diagonal such that the order of the entries remains unchanged. Any hints? Thanks beforehand.