On the geometry of a (q+1)-arc of PG(3, q), q even

In PG(3, q), q = 2n, n & GE; 3, let Ah,q = {(1, t, t2h, t2h+1) | t & ISIN; Fq} ? {(0, 0, 0, 1)}, with gcd(n, h) = 1, be a (q + 1)-arc and let Gh  PGL(2, q) be the stabilizer of Ah,q in PGL(4, q). The Gh-orbits on points, lines and planes of PG(3, q), together with the point-plane incidence matrix with respect to the Gh-orbits on points and planes of PG(3, q) are determined. The point-line incidence matrix with respect to the G1-orbits on points and lines of PG(3, q) is also considered. In particular, for a line B belonging to a given line G1-orbit, say L, the point G1-orbit distribution of B is either explicitly computed or it is shown to depend on the number of elements x in Fq (or in a subset of Fq) such that Trq|2(g(x)) = 0, where g is an Fq-map determined by L.& COPY; 2023 Elsevier B.V. All rights reserved.