I'm trying to find out what is already known about the following setup. 

Let $V$ be an $n$-dimensional vector space over a finite field $F_q$ (I'm mostly interested in the case where $q$ is prime), and let $V_1, V_2, \ldots, V_k$ be proper linear subspaces of $V$ such that the following hold: 

1) $\bigcup\limits_{i=1}^k V_i = V$ (the $V_i$ cover $V$),

2) $V_i\not\subset \bigcup\limits_{j\neq i} V_j$ (the cover doesn't have a subcover), and

3) $\bigcap\limits_{i=1}^k V_i = \{0\}$ (the cover isn't lifting a cover from a quotient of $V$). 


For example, one can always take $\{V_i\}$ to be the collection of all 1-dimensional subspaces of $V$, in which case $k= |\mathbb{P}(V)| = \frac{q^n - 1}{q-1}$. If $n=2$, this is the only choice of $k$ and of $\{V_i\}$. 

For a different example, if $q=2$ and $k=4$, then $n= 4$ is impossible, as one can check. However, $n= 3$ is possible, and there is one cover with $\dim V_1 = \dim V_2 = \dim V_3 = 2$, $\dim V_4 = 1$ and another with $\dim V_1 = \dim V_2 = 2$, $\dim V_3 = \dim V_4 = 1$. If $V$ is spanned by $x, y, z$, then these correspond to $V_1 = <x, y>$, $V_2 = <x, z>$, $V_4 = <x+y+z>$, and $V_3=<y,z>$ or $V_3 = <y+z>$. These are "essentially" the only two choices of $\{V_i\}$, up to relabellings, for $(n,k,q) = (3, 4, 2)$. 

Here are my questions: 

A) For fixed $n$ and $q$, what are the possible values of $k$? What can the arrangements $\{V_i\}$ look like for these $k$? 

B) For fixed $k$ and $q$, what are the possible values of $n$? What can the arrangements $\{V_i\}$ look like for these $n$?