Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with the largest hamming weight such that $\langle v , \frac{1}{r}Mv \rangle$ $=$ $\langle v , v \rangle$.
I am also looking for the rate of growth in the hamming weight as $m \rightarrow \infty$, $\langle v , \frac{1}{r^{m}}M^{\otimes{m}}v \rangle$ $=$ $\langle v , v \rangle$. What are some good mathematical techniques/tools to study this kind of problems?