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.
This is a preview of subscription content,to check access.
Access this article
Similar content being viewed by others
Batten, L.M.: Combinatorics of Finite Geometries, 2nd edn. Cambridge University Press, Cambridge (1997)
Batten, L.M., Beutelspacher, A.: The Theory of Finite Linear Spaces: Combinatorics of Points and Lines. Cambridge University Press, Cambridge (2009)
Betten, A., Betten, D.: Linear spaces with at most 12 points. J. Comb. Des. 7, 119–145 (1999)
Elkies, N.D.: The moduli spaces of the ten \(10_3\)-configurations. Preprint (2016)
Gleason, A.: Finite Fano planes. Am. J. Math. 78(4), 797–807 (1956)
Glynn, D.: Rings of geometries. II. J. Combin. Theory Ser. A 49(1), 26–66 (1988)
Grünbaum, B.: Configurations of Points and Lines. Graduate Studies in Mathematics, 103. American Mathematical Society, Providence (2009)
Hirschfeld, J.W.P., Thas, J.A.: Open problems in finite projective spaces. Finite Fields Appl. 32, 44–81 (2015)
Iampolskaia, A., Skorobogatov, A.N., Sorokin, E.: Formula for the number of [9,3] MDS codes. IEEE Trans. Inf. Theory 41(6), 1667–1671 (1995)
Kaplan, N., Kimport, S., Lawrence, R., Peilen, L., Weinreich, M.: The number of \(10\)-arcs in the projective plane is not quasipolynomial (in preparation) (2017)
Lawrence, R., Weinreich, M.: Counting 10-arcs in the projective plane over a finite field. http://rachellawrence.github.io/TenArcs/
McKay, B.D., Piperno, A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2013)
Rolland, R., Skorobogatov, A.N.: Dénombrement des configurations dans le plan projectif, In: Proceedings of Arithmetic, Geometry, and Coding Theory (Luminy 1993), 199–207, de Gruyter, Berlin (1996)
Sturmfels, B.: Computational algebraic geometry of projective configurations. J. Symb. Comput. 11, 595–618 (1991)
Tait, M.: On a problem of Neumann. arXiv:1509.06587 (2015), p. 4.
The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2016, Sequence A001200
Weibel, C.: Survey of non-Desarguesian planes. Not. Am. Math. Soc. 54(10), 1294–1303 (2007)
About this article
Cite this article
Kaplan, N., Kimport, S., Lawrence, R. et al. Counting arcs in projective planes via Glynn’s algorithm. J. Geom. 108, 1013–1029 (2017). https://doi.org/10.1007/s00022-017-0391-1