On next-to-minimum size blocking sets of external lines to a nondegenerate quadric in $\operatorname{PG}(3,q)$

Consider a hyperbolic or an elliptic quadric $Q^\epsilon(3,q)$, $\epsilon \in \{ +,- \}$, in $\operatorname{PG}(3,q)$ and let $\mathcal{E}^\epsilon$ denote the set of all lines of $\operatorname{PG}(3,q)$ that are external with respect to $Q^\epsilon(3,q)$. If $\pi$ is a (tangent or secant) plane of $\operatorname{PG}(3,q)$, then $\pi \setminus Q^\epsilon(3,q)$ is an $\mathcal{E}^\epsilon$-blocking set which is minimal, except when $(q,\epsilon) = (2,+)$ and $\pi$ is a secant plane. In this way, we obtain two families of (minimal) blocking sets of sizes $m_1^\epsilon$ and $m_2^\epsilon$, with $(m_1^+,m_2^+)=(q^2-q,q^2)$ and $(m_1^-,m_2^-)=(q^2,q^2+q)$, where those of size $m_1^\epsilon$ are also the minimum size $\mathcal{E}^\epsilon$-blocking sets. Motivated by the search for new (families of) minimal $\mathcal{E}^\epsilon$-blocking sets with sizes in the open interval $]m_1^\epsilon,m_2^\epsilon[$, we determine here all $\mathcal{E}^\epsilon$-blocking sets of size $m_1^\epsilon+1$ in $\operatorname{PG}(3,q)$ in the case $q \in \{ 2,3 \}$.

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