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Counting arcs in projective planes via Glynn’s algorithm

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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.

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Correspondence to Nathan Kaplan.

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Kaplan, N., Kimport, S., Lawrence, R. et al. Counting arcs in projective planes via Glynn’s algorithm. J. Geom. 108, 1013–1029 (2017).

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