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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.

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

Apoorva Khare, Vector spaces as unions of proper subspaces. Linear Algebra Appl. 431 (2009), no. 9, 1681–1686.

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  • $\begingroup$ Yes, that answers my question. Thank you so much for the much-needed pointer! $\endgroup$ – Mohammed Karmoose Apr 21 '17 at 17:54
  • $\begingroup$ You're welcome. The article was a discovery for me as well. The guy's good, and the paper is sharp and well written. $\endgroup$ – Andreas Caranti Apr 21 '17 at 19:17

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