An algorithm for counting arcs in higher-dimensional projective space

Abstract

An n-arc in (k−1)-dimensional projective space is a set of n points so that no k lie on a hyperplane. In 1988, Glynn gave a formula to count n-arcs in the projective plane in terms of simpler combinatorial objects called superfigurations. Several authors have used this formula to count n-arcs in the projective plane for n≤10. In this paper, we determine a formula to count n-arcs in projective 3-space. We then use this formula to give exact expressions for the number of n-arcs in P3(Fq) for n≤7, which are polynomial in q for n≤6 and quasipolynomial in q for n=7. Lastly, we generalize to higher-dimensional projective space.