# Find the minimum number of subspaces of max. dimension $k$ to cover finite field $\mathbb{F}_q^n$

My question is the following: assume a vector space of dimension $n$ over a finite field of size $q$, that is $\mathbb{F}_q^n$. Let $1 < k < n$. I would like to find the minimum number of subspaces of $\mathbb{F}_q^n$ of dimension at most $k$ that could cover all elements of $\mathbb{F}_q^n$. To be more concrete, let $\mathcal{V} = \{V_1, ... , V_T \}$ be a set of subspaces $V_i \subseteq \mathbb{F}_q^n$ where $dim(V_i) \leq k$ for all $i = 1, ... , T$ and $V_1 \cup ... \cup V_T = \mathbb{F}_q^n$. Then I would like to find the minimum possible value of $T$ to be able to construct such a set $\mathcal{V}$.

A very similar question is the one answered in this paper. However, they allow the use of $k$-flats, which could be cosets of subspaces. In my question, I'm more constrained in the sense that I only allow subspaces.

• Do you really mean $V_1 + \dots + V_T = \mathbb{F}_q^n$ or perhaps $V_1 \cup \dots \cup V_T = \mathbb{F}_q^n$? – Andreas Caranti Apr 21 '17 at 11:29
• You are right, I meant the union. I'll edit. – Mohammed Karmoose Apr 21 '17 at 15:41