Counting arcs in projective planes via Glynn’s algorithm

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 ≤ 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.

Seven points in general linear position

“…the work of Glynn [11] and continued in the work of Iampolskaia, Skorobogatov and Sorokin [13] and Kaplan et al [14]. In particular, the number |C 2,m (F q )| of m-tuples of points in general linear position in the projective plane over the finite field F q with q elements is known for all q and m < 10 through these works.…”

On the number of $k$-gons in finite projective planes

“…Furthermore, in [15], Glynn computes an expression for the number of 7-arcs in any finite projective plane, and using this expression deduces that there do not exist projective planes of order 6, as evaluating the formula at 6 yields a negative value. Glynn's work counting k-arcs was recently extended by Kaplan, Kimport, Lawrence, Peilen and Weinreich [20] who determined an expression for the number of 9-arcs in an arbitrary projective plane. It is worth mentioning that for k = 7, 8, 9, the formula for the number of k-arcs depends on more than just k and the order of the plane.…”

An algorithm for counting arcs in higher-dimensional projective space

“…, g N −1 so that f = g i whenever q ≡ i (mod N). The function C n,3 (Π) is quasipolynomial when n ∈ {7, 8, 9} [5,10,12]. Iampolskaia, Skorobogatov, and Sorokin [10] count [9,3] MDS codes and derive their formula for C 9,3 (P 2 (F q )) as a corollary.…”