Incidence matrices for the class \({\mathcal{O}}_6\) of lines external to the twisted cubic in \(\mathrm{PG}(3,q)\) (Q6115813)

A twisted cubic is a normal rational curve \(\mathcal{C}\) in the \(3\)-dimensional projective space \(\mathrm{PG}(3, q)\), represented in standard form as \(\mathcal{C} = \{(t^3, t^2, t, 1) : t \in \mathbb{F}_q\} \cup \{(1, 0, 0, 0)\}\). It exhibits numerous interesting properties and finds a wide range of applications, including finite geometry, coding theory, and cryptography. The stabilizer \(G_q\simeq\mathrm{PGL}(2,q)\) of the twisted cubic acts on points, lines and planes of \(\mathrm{PG}(3,q)\). On points and planes \(G_q\) acts with \(5\) orbits (see [\textit{J. W. P. Hirschfeld}, Finite projective spaces of three dimensions. Oxford: Oxford University Press (1985; Zbl 0574.51001)]), whereas a classification of the orbits on lines its still to be obtained in the general case (the number of orbits on lines has been computed in [\textit{M. Ceria} and \textit{F. Pavese}, Finite Fields Appl. 84, Article ID 102098, 16 p. (2022; Zbl 1498.51008)] for \(q\) even). The lines can be divided in \(6\) classes, each union of \(G_q\)-orbits. The class \(\mathcal{O}_6\) of lines external to the twisted cubic is the only class still to be decomposed as union of \(G_q\)-orbits. Recently, [\textit{A. A. Davydov} et al., Mediterr. J. Math. 20, No. 3, Paper No. 160, 25 p. (2023; Zbl 1519.51006)], the authors of this paper found a family \(\mathfrak{F}\) of lines of \(\mathcal{O}_6\). The main result of the paper is to provide the plane-line and point-line incidence matrices for this family \(\mathfrak{F}\) of lines.