Dual of blocking sets in finite geometry

Question

Let $V$ be an $n$-dimensional vector space over the finite field of cardinality $q$ and let $W_1,\ldots,W_m$ be hyperplanes of $V$ such that $$V=\bigcup_{i=1}^mW_i \,\,\hbox{ and }\,\,0=\bigcap_{i=1}^mW_i.$$

I am interested in a lower bound for $m$ in terms of $n$ and $q$. For instance, when $n=2$, we have $m=q+1$.

The collection $\{W_1,\ldots,W_m\}$ of hyperplanes with the conditions above seems to be the "dual" of a blocking set (for projective plane). A blocking set $S$ in a projective plane is a subset of points with the property that every line intersect non-trivially $S$, and $S$ does not contain a line. So, possibly my question has already an answer in the literature. But I was not able to find it, besides for planes.

Answer 1

The trivial bound $m=q+1$ mentioned in the case where $n=2$ is actually true for all $n$ by simple counting: a set of $k$ hyperplanes contains at most $kq^{n-1}+q^{n-2}+\ldots+1$ points, since the "first" hyperplane has $q^{n-1}+\ldots+1$ points, then each subsequent hyperplane meets the "first" hyperplane in $q^{n-2}+\ldots+1$ points, and thus adds at most $q^{n-1}$ new points to the union. But this bound clearly cannot be met without violating the condition that the intersection of all hyperplanes be trivial.

In the plane ($n=2$), the actual lower bound on a blocking set is $q+\sqrt{q}+1$, due to Bruen [Bruen, A., Blocking sets finite projective planes, SIAM J. Appl. Math. 21 (1971), 380–892]. The lower bound is met if and only if the blocking set is a Baer subplane (i.e., a subplane of order $\sqrt{q}$), which occurs only if $q$ is square. There is a substantial literature regarding the situation when $q$ is not square; Hirschfeld [Hirschfeld, J W P, 'Blocking sets', Projective Geometries over Finite Fields (Oxford, 1998; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198502951.003.0013, accessed 10 Nov. 2024.] provides an entire chapter surveying results in this direction.

The situation in higher dimensions is not nearly as well studied. A good reference which addresses some of the issues is Sziklai [P. Sziklai. On small blocking sets and their linearity. J. Combin. Theory Ser. A, 115:1167–1182, 2008.]. In particular, this article contains some relatively small examples of blocking sets which are based on subgeometries, but I do not believe there is any sort of general proof of a lower bound here.