International Journal of Nonlinear Analysis and Applications

The smallest size of the arc of degree three in a projective plane of order sixteen

Document Type : Research Paper

Authors

Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq

Abstract

 An $(n;3)$-arc in projective plane $PG(2,q)$ of size n and degree three is a set of n points such that no four of them collinear but some three of them are collinear.An $(n;r)$-arc is said to be complete if it is not contained in $(n+1;r)$-arc.  The aim of this paper is to construct the projectively distinct$(n;3)$-arcs in $PG(2,16)$, determined the smallest complete arc in $PG(2,16)$ then the stabilizer group of these arcs are established and we have identified the group with which it's isomorphic.

Keywords

[1] S. Alabdulla and J.W.P. Hirschfeld, A new bound for the smallest complete (k, n)-arc in PG(2, q), Des. Codes Crypt. 87(2-3) (2019) 679–683.
[2] N.A.M. Al-Seraji, The group action on the finite projective planes of orders 53,61,64, J. Discrete Math. Sci. Crypt. 23(8) (2020) 1573–1582.
[3] N.A.M. Al-Seraji, A. Bahkeet and Z. Sadiq, Study of Orbits on the finite projective plane, J. Interdiscip. Math. 23(6) (2020) 1187–1195.
[4] S. Ball and J.W.P. Hirschfeld, Bounds on (n; r)-arcs and Their Application to Linear Codes, Department of Mathematics, University of Sussex, Brighton BNI 9RF.UK., 2005.
[5] A. Barlotti, Sui (K; n)-archi di un piao linear finite, Boll. Un. Mat. Ttal. 11(3) (1956) 553–556.
[6] K. Coolsaet, The complete arcs of PG(2,31), J. Combin. Des. 23 (2015) 522–533.
[7] M. Dillon, Geometry Through History, Springer, 2018.
[8] Gap Group, GAP, Reference manual URL, 2021, http//www.gap-system.org.
[9] Z.S. Hamed and J.W. Hirschfeld, A Complete (48,4) in the Projective Plane Over the Field of Order Seventeen, Department of Mathematics, School of Mathematical and Physical Science, Brighton, UK., 2018.
[10] J.W.P. Hirschfeld, Finite Projective Space Of Three Dimensions, Oxford University Press, Oxford, 1985.
[11] J.W.P. Hirschfeld, Projective Geometries Over Finite Feilds, 2nd Edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.
[12] J.W.P. Hirschfeld and J.A. Thas, General Galois Gemetries, Oxford University Press, Oxford, 1991.
[13] J.W.P. Hirschfeld and J.A.Thas, Hermitian varieties, General Galois Geometries. Springer, London, 2016.
[14] J.W.P. Hirschfield and J.F. Voloch, Group-arcs of prime order on cubic curves, Finite Geom. Comb. 191 (2015) 177–185.
[15] G. Kiss and T. Szonyi, Finite Geometry, Boca Raton, CRC Press, 2019.
[16] S. Marcugini, A. Milani and F. Pambianco, Classification of the (n; 3)-arcs in PG(2,7), J. Geom. 80(2004) 197–184.
[17] E.V.D. Pichanick and J.W.P. Hirschfeld, Bounded for arcs of arbitrary degree in finite Desarguesian planes, J. Comb. Des. 24(4) (2016) 184–196.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 3749-3764
  • Receive Date: 10 June 2021
  • Revise Date: 24 September 2021
  • Accept Date: 11 October 2021