Let $\mathbb{F}_q$ be a finite field with $q$ elements. Given $p_1, p_2 , \cdots , p_n $, a number of points (not necessarily distinct) of $\mathbb{F}_q^m$ chosen randomly from a uniform distribution.
Let $V$ be a subspace of $\mathbb{F}_q^m$ of dimension $m-1$ and let $N_V$ be the number of the points $p_1 , ... , p_n$ that lie on $V$. It is easy to see that this is distributed as $N_V \sim B(n ,1/q )$ , the binomial distribution with $n$ trials and success probability $1/q$.
However, I am interested in $M$, the maximum of $N_V$ over all $m-1$-dimensional subspaces of $\mathbb{F}_q^m$.
The number of $m-1$-dimensional subspaces is $(q^m-1)/(q-1)$, but the distribution of $M$ is not just the distribution of $(q^m-1)/(q-1)$ independent binomial distributions, because the $N_V$ are not independent.
What is the distribution of $M$? Maybe the distribution of $M'$, the maximum of $(q^m-1)/(q-1)$ independent binomial distributions is a good approximation?
EDIT: I am mainly interested in the case $n \ge m$ since in the other case $M = n$
I manually calculated the probabilities for $q=2 , m = 2$ and $n =4$
\begin{array}{ c c c } X & P(M = X) & P(M' = X) \\ \hline 0 & 0 & 0 \\ 1 & 0 & 0.03 \\ 2 & 0.305 & 0.294 \\ 3 & 0.516 & 0.499 \\ 4 & 0.180 & 0.176 \\ \end{array}
This seems to suggest that the maximum of independent binomial distributions is a good approximation.
Thanks in advance.