Counting Arcs in Projective Planes via Glynn's Algorithm

TL;DR

This paper derives a general formula for counting 9-arcs in any finite projective plane, including non-Desarguesian ones, using a new implementation of Glynn's algorithm, extending known results for smaller arcs.

Contribution

It provides a new, general formula for 9-arcs in all finite projective planes, expanding previous work limited to specific cases and sizes.

Findings

Derived a formula for 9-arcs in any projective plane of order q.

Implemented Glynn's algorithm to facilitate counting larger arcs.

Discussed implications for counting larger arcs beyond 9.

Abstract

An $n$-arc in a projective plane is a collection of $n$ distinct points in the plane, no three of which lie on a line. Formulas counting the number of $n$-arcs in any finite projective plane of order $q$ are known for $n \le 8$. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted $9$-arcs in the projective plane over a finite field of order $q$ and showed that this count is a quasipolynomial function of $q$. We present a formula for the number of $9$-arcs in any projective plane of order $q$, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as a special case. We obtain our formula from a new implementation of an algorithm due to Glynn; we give details of our implementation and discuss its consequences for larger arcs.