Advertisement

Designs, Codes and Cryptography

, Volume 55, Issue 2–3, pp 285–296 | Cite as

Infinite family of large complete arcs in PG(2, q n ), with q odd and n > 1 odd

  • Gábor Korchmáros
  • Nicola Pace
Article

Abstract

For q odd and n > 1 odd, a new infinite family of large complete arcs K′ in PG(2, q n ) is constructed from complete arcs K in PG(2, q) which have the following property with respect to an irreducible conic \({\mathcal{C}}\) in PG(2, q): all the points of K not in \({\mathcal{C}}\) are all internal or all external points to \({\mathcal{C}}\) according as q ≡ 1 (mod 4) or q ≡ 3 (mod 4).

Keywords

Arc Projective plane Collineation group 

Mathematics Subject Classification (2000)

51E21 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abatangelo V., Fisher J.C., Korchmáros G., Larato B.: On the mutual position of two irreducible conics in PG(2, q), q odd, submitted.Google Scholar
  2. 2.
    Boros E., Szőnyi T.: On the sharpness of a theorem of B. Segre. Combinatorica 6, 261–268 (1986)zbMATHCrossRefGoogle Scholar
  3. 3.
    Cossidente A.: A new proof of the existence of (q 2q + 1)-arcs in PG(2, q 2. J. Geom. 53, 37–40 (1995); correction 59, 32–33 (1997).Google Scholar
  4. 4.
    Cossidente A., Korchmáros G.: The Hermitian function field arising from a cyclic arc in a Galois plane. Geometry, Combinatorial Designs and Related Structures, London Mathematical Society Lecture Note Series, vol. 245, pp. 63–68. Cambridge University Press, Cambridge (1997).Google Scholar
  5. 5.
    Cossidente A., Korchmáros G.: The algebraic envelope associated to a complete arc. Rend. Circ. Mat. Palermo Suppl. 51, 9–24 (1998)Google Scholar
  6. 6.
    Ebert G.: Partitioning projective geometries into caps. Can. J. Math. 37, 1163–1175 (1985)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Fisher J.C., Hirschfeld J.W.P., Thas J.A.: Complete arcs in planes of square order. Combinatorics ’84. North-Holland Mathematics Studies, vol. 123, pp. 243–250. North-Holland, Amsterdam (1986).Google Scholar
  8. 8.
    Giulietti M.: On plane arcs contained in cubic curves. Finite Fields Appl. 8, 69–90 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hirschfeld J.W.P., Korchmáros G.: Embedding an arc into a conic in a finite plane. Finite Fields Appl. 2, 274–292 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hirschfeld J.W.P., Korchmáros G.: On the number of rational points on an algebraic curve over a finite field. Bull. Belg. Math. Soc. Simon Stevin. 5, 313–340 (1998)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Hirschfeld J.W.P., Korchmáros G., Torres F.: Algebraic Curves Over a Finite Field, pp. xx+696. Princeton University Press, Princeton and Oxford (2008).Google Scholar
  12. 12.
    Kestenband B.: Unital intersections in finite projective planes. Geom. Dedicata 11, 107–117 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Korchmáros G.: A combinatorial characterisation of the dihedral subgroups of order 2(p r + 1) of PGL (2, p r). Geom. Dedicata 9, 381–384 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Korchmáros G., Sonnino A.: On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order. J. Combin. Designs (2009). doi: 10.1002/jcd.20220.
  15. 15.
    Valentini R.C., Madan M.L.: A Hauptsatz of L.E. Dickson and Artin–Schreier extensions. J. Reine Angew. Math. 318, 156–177 (1980)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Voloch J.F.: On the completeness of certain plane arcs. Eur. J. Combin. 8, 453–456 (1987)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Voloch J.F.: On the completeness of certain plane arcs II. Eur. J. Combin. 11, 491–496 (1990)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Voloch J.F.: Arcs in projective planes over prime fields. J. Geom. 38, 198–200 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Voloch J.F. et al.: Complete arcs in Galois planes of non-square order. In: Hirschfeld, J.W.P. (eds) Advances in Finite Geometries and Designs, Isle of Thorns 1990., pp. 401–405. Oxford University Press, Oxford (1991)Google Scholar
  20. 20.
    Waterhouse W.C.: Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. 2, 521–560 (1969)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità della BasilicataPotenzaItaly
  2. 2.Department of Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA

Personalised recommendations