Twisted cubic and plane-line incidence matrix in PG(3, q)

The point-plane, the point-line, and the plane-line incidence matrices of PG(3, q) are of interest in combinatorics, finite geometry, graph theory and group theory. Some of the properties of these matrices and their submatrices are related with the interplay of orbits of points, lines and planes under the action of subgroups of PGL(4, q). A remarkable particular case is the subgroup G congruent to PGL(2, q), viewed as the stabilizer group of the twisted cubic C. For this case, the study of the point-plane incidence matrix, initiated by D. Bartoli and the present authors, has attracted attention as being related to submatrices with useful applications in coding theory for the construction of multiple covering codes. In this paper, we extend our investigation to the plane-line incidence matrix apart from just one class of the line orbits, named O-6 in the literature. For all q >= 2, in each submatrix, the numbers of lines in any plane and planes through any line are obtained. As a tool for the present investigation, we use submatrices of incidences arising from orbits of planes and unions of line orbits, including O-6. For each such submatrix we determine the total number of lines from the union in any plane and the average number of planes from the orbit through any line of the union.