Blocking Planes by Lines in $\operatorname{PG}(n,q)$

TL;DR

This work investigates the minimum size $f(n,q)$ of a $(2,1)$-blocking set in $PG(n,q)$, i.e., the smallest set of lines meeting every plane. It develops a refined recursive construction that builds blocking sets from smaller dimensional pieces, yielding improved upper bounds on $f(n,q)$ and identifying an optimal recursion strategy where $n-k$ tends to be a power of two. Lower bounds are sharpened via the $q$-analogue Schönheim bound and standard equations, and the density $ ho(n,q)=f(n,q)/|inom{n+1}{2}_q|$ is analyzed, including asymptotic behavior and limits. The results connect blocking sets to $q$-covering and $q$-Turán designs and to subspace designs, with implications for Grassmannian codes and network coding, and they outline open problems about precise asymptotics and general $(s,t)$-blocking regimes.

Abstract

In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces $\operatorname{PG}(n,q)$ such that every plane is incident with at least one line of the set. This is the first main open problem concerning the minimum size of $(s,t)$-blocking sets in $\operatorname{PG}(n,q)$, where we set $s=2$ and $t=1$. In $\operatorname{PG}(n,q)$, an $(s,t)$-blocking set refers to a set of $t$-spaces such that each $s$-space is incident with at least one chosen $t$-space. This is a notoriously difficult problem, as it is equivalent to determining the size of certain $q$-Turán designs and $q$-covering designs. We present an improvement on the upper bounds of Etzion and of Metsch via a refined scheme for a recursive construction, which in fact enables improvement in the general case as well.

Figures (2)

• Figure 2.1: The lattice $\mathcal{G}_{\bullet}(X)$ of all subspaces of $X$ (see \ref{['def:grassmann']}). For a fixed $K\in\mathcal{G}_{\bullet}(X)$, elements smaller than $K$ can be identified with $\mathcal{G}_{\bullet}(K)$ (blue part), while elements bigger than $K$ can be identified with $\mathcal{G}_{\bullet}(X/K)$ (red part) (see \ref{['def:quotientGeometry']}). The maps $\pi_K$ (mapping to the quotient space $X/K$) and $\rho_K$ (mapping to the subspace $K$) are indicated (see \ref{['def:shortExactSequence']}). In fact, $\langle K,Y \rangle$ is the least upper bound of $K$ and $Y$, and $K\cap Y$ is the greatest lower bound of $K$ and $Y$ in the lattice $\mathcal{G}_{\bullet}(X)$. The dimensional formula of \ref{['eq:dimension_Pi_Rho']} translates to the fact that the length of the vertical segment on the left (black) equals the sum of the lengths of the vertical segments on the right (blue and red) -- assuming that subspaces of the same dimension are represented by points (of the lattice above) lying on a horizontal line, and that these lines are equidistant and are increasing with the dimension.
• Figure 3.1: The main idea of the proof of \ref{['partial']} is indicated in this figure. Start from a subspace $S\in\mathcal{S}$ to be blocked. Then $S_1\coloneqq\rho_K(S)\in\mathcal{S}_{K}$ is blocked by some $T_1\in\mathcal{B}_{K}$, and $S_2\coloneqq\pi_K(S)\in\mathcal{S}_{X/K}$ is blocked by some $T_2\in\mathcal{B}_{X/K}$. By \ref{['lem:T1T2']}, there exists $T\in\mathcal{G}_{\bullet}(X)$ such that $\rho_K(T)=T_1$ and $\pi_K(T)=T_2$. Finally, one can show that $T$ is actually from $\mathcal{B}$ and blocks $S$. (Here, thick horizontal dashed lines indicate subsets consisting of subspaces of given dimension. Shaded regions between these lines indicate the connection between partial blocking sets and the sets they block.)

Theorems & Definitions (94)

• Definition 1.1: $t$-spread
• Theorem 1.2: Beutelspacher and Ueberberg, beutel
• Theorem 1.3: Eisfeld and Metsch, eis, Metsch
• Definition 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7: Etzion, Vardy, Etz2
• Proposition 1.8: Lower bound
• Theorem 1.9: Upper bound
• Definition 2.1