In $\mathrm{ PG}(3, q)$, $q = 2^n$, $n \ge 3$, let ${ \cal A} = \{ (1,t,t^{ 2^h},t^{ 2^h+1}) \mid t \in \mathbb{ F}_q\} \cup \{ (0,0,0,1)\}$, with $\mathrm{ gcd}(n,h) = 1$, be a $(q+1)$-arc and let $G_h \simeq \mathrm{ PGL}(2, q)$ be the stabilizer of $\cal A$ in $\mathrm{ PGL}(4, q)$. The $G_h$-orbits on points, lines and planes of $\mathrm{ PG}(3, q)$, together with the point-plane incidence matrix with respect to the $G_h$-orbits on points and planes of $\mathrm{ PG}(3, q)$ are determined. The point-line incidence matrix with respect to the $G_1$-orbits on points and lines of $\mathrm{ PG}(3, q)$ is also considered. In particular, for a line $\ell$ belonging to a given line $G_1$-orbits, say $\cal L$, the point $G_1$-orbit distribution of $\ell$ is either explicitly computed or it is shown to depend on the number of elements $x$ in $\mathbb{ F}_q$ (or in a subset of $\mathbb{ F}_q$) such that $\mathrm{ Tr}_{ q|2}(g(x)) = 0$, where $g$ is an $\mathbb{ F}_q$-map determined by $\cal L$.
Categories: math.CO