Question

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?

Answer 1

Perhaps I misunderstand, but if $v$ is all ones (and you can't get a bigger Hamming weight than that) then $Mv$ is all $r$, so $(1/r)Mv$ is all ones again, so $(1/r)Mv=v$ and the inner product of $v$ with $(1/r)Mv$ is the inner product of $v$ with $v$.