Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ whose entries are also chosen independently and uniformly from $\{0,1\}$. Let $N = Mv$ where multiplication is performed over the reals.

The matrices $M$ and $N$ are discrete random variables. Recall that the Shannon entropy for a discrete random variable $Z$ is $H(Z) = -\sum_z P(Z=z)\log_2{P(Z=z)}$. In the case where $P(Z=z)=0$ for some values $z$, the corresponding term in the sum is taken to be $0$.

We therefore know that the (base $2$) Shannon entropy $H(M) = H(v) = n$. The fact that $H(M) = n$ is a direct result of the fact that the entire matrix is defined by its first row.

If $m = \lfloor 10n/\ln{n} \rfloor$ then I would like to make the following conjecture.

Conjecture:For all sufficiently large $n$, $H(N) \geq n/10$.

The value $10$ is chosen somewhat arbitrarily to be a sufficiently large constant.

Is this a known problem and/or can anyone see a way to approach it? Is it in fact true?