**What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line?**

Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0, y = 0$ and $z = 0$, forms such a set. If we denote by $s(q)$ the smallest possible size, then this gives us $$s(q) \leq 3q^2 - 3q + 1$$ for all $q$. 

Since any two points of $\mathbb{F}_2^3$ form a line, we get $s(2) = 2^3 - 1 = 7 = 12 - 6 + 1$. For $q = 3$ we can do better. In $\mathbb{F}_3^3$ the complement of such a set is a **cap**, i.e., a set of points no three of which are collinear. We know that the largest size of a cap in $\mathbb{F}_3^3$ is $9$ (corresponding to a quadric), and hence $s(3) = 18 < 27 - 9 + 1$. (side note: the problem of finding such a blocking set in $\mathbb{F}_3^n$ is thus equivalent to the famous cap set problem. See the survey article by Bierbrauer and Edel, [large caps in projective Galois spaces][1] and the paper by Bateman and Katz, [new bounds on cap sets][2]) 

**Question 1:** Can we improve the upper bound in general?

We can also give a lower bound on $s(q)$. Jamison/Brouwer-Schrijver proved using the polynomial method that the smallest possible size of a blocking set in $\mathbb{F}_q^2$ is $2q - 1$. See [this][3], [this][4], [this][5] and [this][6] for various proofs of their result. Now take any $q$ parallel affine planes in $\mathbb{F}_q^3$, then the intersection of a blocking set with these hyperplanes must have size at least $2q - 1$, and hence $$2q^2 - q \leq s(q).$$ 

**Question 2:** Can we improve this lower bound in general?

The Jamison/Brouwer-Schrijver result gives us another way of constructing a blocking set of size $3q^2 - 3q + 1$. Again take $q$ parallel hyperplanes $H_1, \dots, H_q$. Let $B_2, \dots, B_{q}$ be blocking sets of size $2q - 1$ in $H_2, \dots, H_{q}$. Then $B = H_1 \cup B_2 \cup \dots \cup B_{q}$ is a blocking set of size $(q-1)(2q-1) + q^2 = 3q^2 - 3q + 1$. 

Note that the problem of determining $s(q)$ is trivial for projective spaces. It's a classical result that a line blocking set in $PG(3,q)$ has size at least $1 + q + q^2$ with equality if and only if it is a hyperplane. See Chapter 3 of [current research topics in Galois geometry][7] for a recent survey on projective blocking sets. 

**Edit 1:** After Douglas Zare's improvement we have $s(q) \geq 2q^2 - 1$ for all $q$ and $s(q) \leq 3q^2 - 3q$ for $q \geq 3$. Can this be improved further?


  [1]: https://www.mathi.uni-heidelberg.de/~yves/Papers/CapSurvey.pdf
  [2]: http://www.ams.org/journals/jams/2012-25-02/S0894-0347-2011-00725-X/
  [3]: http://www.sciencedirect.com/science/article/pii/0097316577900012
  [4]: http://www.sciencedirect.com/science/article/pii/0097316578900134
  [5]: http://www.tau.ac.il/~nogaa/PDFS/tools1.pdf
  [6]: http://arxiv.org/abs/1508.06020
  [7]: https://www.novapublishers.com/catalog/product_info.php?products_id=21439