An algorithm for counting arcs in higherdimensional projective space
Abstract
An $n$ arc in $(k1)$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 $n \le 10$. In this paper, we determine a formula to count $n$arcs in projective 3space. We then use this formula to give exact expressions for the number of $n$arcs in $\mathbb{P}^3(\mathbb{F}_q)$ for $n \le 7$, which are polynomial in $q$ for $n \le 6$ and quasipolynomial in $q$ for $n=7$. Lastly, we generalize to higherdimensional projective space.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.01024
 Bibcode:
 2021arXiv210801024I
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 20 pages