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 $n \le 10$. 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 $\mathbb{P}^3(\mathbb{F}_q)$ for $n \le 7$, which are polynomial in $q$ for $n \le 6$ and quasipolynomial in $q$ for $n=7$. Lastly, we generalize to higher-dimensional projective space.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2021
- DOI:
- 10.48550/arXiv.2108.01024
- arXiv:
- arXiv:2108.01024
- Bibcode:
- 2021arXiv210801024I
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- Revised based on referee suggestions. Accepted to Finite Fields and Their Applications