Minimum size blocking sets of certain line sets with respect to an elliptic quadric in $PG(3,q)$

Abstract

For a given nonempty subset $\mathcal{L}$ of the line set of $\PG(3,q)$, a set $X$ of points of $\PG(3,q)$ is called an $\mathcal{L}$-blocking set if each line in $\mathcal{L}$ contains at least one point of $X$. Consider an elliptic quadric $Q^-(3,q)$ in $\PG(3,q)$. Let $\mathcal{E}$ (respectively, $\mathcal{T}, \mathcal{S}$) denote the set of all lines of $\PG(3,q)$ which meet $Q^-(3,q)$ in $0$ (respectively, $1,2$) points. In this paper, we characterize the minimum size $\mathcal{L}$-blocking sets in $\PG(3,q)$, where $\mathcal{L}$ is one of the line sets $\mathcal{S}$, $\mathcal{E}\cup \mathcal{S}$, and $\mathcal{T}\cup \mathcal{S}$.