Advertisement

Journal of Geometry

, 109:14 | Cite as

Non-existence of sets of type \(\mathbf (0, 1, 2 , {\varvec{n}}_{{\varvec{d}}})_{{\varvec{d}}}\) in PG(\({\varvec{r,q}}\)) with \(\mathbf 3 \le {\varvec{d}}\le {\varvec{r}}-\mathbf 1 \) and \({\varvec{r}}\ge \mathbf 4 \)

  • Mauro Zannetti
Article
  • 34 Downloads

Abstract

This paper deals with sets of type \((0,1,2,n_{d})_{d}\) in PG(rq), \(1\le d\le r-1\). The non-existence of sets of type \((0,1,2,n_{d})_{d}\), \(3\le d\le r-1\) in PG(rq) with \(r\ge 4\) is proved.

Keywords

Sets of type \((0, 1, 2, n_{d})_{d}\) Four-intersection sets Sets with few intersection numbers 

Mathematics Subject Classification

51E20 

References

  1. 1.
    Ball, S., Blokhuis, A.: The classification of maximal arcs in small Desarguesian planes. Bull. Belg. Math. Soc. Simon Stevin 9, 433–445 (2002)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bichara, A.: On \(k\) -sets of class \([0,1,2, n]_{2}\) in PG(\(r, q\)), Finite geometries and designs (Proceedings of the Conference on Chelwood Gate, 1980). London Mathematical Society Lecture Notes Series, vol. 49 (1981), pp. 31–39 (1980)Google Scholar
  3. 3.
    De Clerck, F., De Feyter, N.: A characterization of the sets of internal and external points of a conic. Eur. J. Comb. 28, 1910–1921 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Durante, N.: On sets with few intersection numbers in finite projective and affine spaces. Electron. J. Comb. 21(4), 4–13 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Esposito, R.: Sui \(k\) -insiemi di tipo \((0,1,m_{d})_{d}\) di uno spazio di Galois \(S_{r,q}\) \((2\le d < r, r \ge 3)\). Riv. Mat. Univ. Parma (4) 3, 131–140 (1977)Google Scholar
  6. 6.
    Hirschfeld, J.W.P., Thas, J.A.: General Galois Geometries. Clarendon Press, Oxford (1991)zbMATHGoogle Scholar
  7. 7.
    Innamorati, S.: A characterization of the set of lines either external to or secant to an ovoid in PG (\(3, q\)). Australas. J. Comb. 49, 159–163 (2011)Google Scholar
  8. 8.
    Innamorati, S., Tondini, D.: The yin-yang structure of the affine plane of order four. Ars Comb. 110, 193–197 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Innamorati, S., Zannetti, M.: A characterization of the planes secant to a non-singular quadric in PG (4, q). Ars Comb. 102, 435–445 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Innamorati, S., Zannetti, M., Zuanni, F.: A characterization of the lines external to a hyperbolic quadric in PG (3, q). Ars Comb. 103, 3–11 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Innamorati, S., Zannetti, M., Zuanni, F.: A characterization of the non-Baer lines of a Hermitian surface. J. Geom. 103, 285–291 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Innamorati, S., Zannetti, M., Zuanni, F.: A combinatorial characterization of the Hermitian surface. Discrete Math. 313, 1496–1499 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Innamorati, S., Zannetti, M., Zuanni, F.: On two-character \((q ^{7}+q ^{5}+q ^{2}+1)\) -sets in \(\text{PG}(4, q ^{2})\). J. Geom. 106(6), 287–296 (2015)Google Scholar
  14. 14.
    Innamorati, S., Zuanni, F.: In AG (\(3, q\)any \(q ^{2}\) -set of class \([1, m, n]_{2}\) is a cap. Discrete Math. 338, 863–865 (2015)Google Scholar
  15. 15.
    Innamorati, S., Zuanni, F.: In \(\text{ AG }(3, q)\) any \(q ^{2}\) -set of class \([0, m, n]_{2}\) containing a line is a cylinder. J. Geom. 108, 457–462 (2017)Google Scholar
  16. 16.
    Napolitano, V.: On \((q ^{2}+q+1)\) -sets of class \([1, m, n]_{2}\) in \(\text{ PG }(3, q)\). J. Geom. 105, 448–455 (2014)Google Scholar
  17. 17.
    Tallini Scafati, M.: Sui \(k\) -insiemi di uno spazio di Galois \(S_{r,q}\) a due soli caratteri nella dimensione \(d\). Atti Accad. Naz. Lincei 40, 782–788 (1976)Google Scholar
  18. 18.
    Thas, J.A.: A combinatorial problem. Geom. Dedic. 1, 236–240 (1973)Google Scholar
  19. 19.
    Ueberberg, J.: On regular \(\{v, n\}\) -arcs in finite projective spaces. J. Comb. Des. 1, 395–409 (1993)Google Scholar
  20. 20.
    Zannetti, M.: A combinatorial characterization of parabolic quadrics. J. Discrete Math. Sci Cryptogr. 12, 707–715 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zannetti, M.: A characterization of the external lines of a hyperoval cone in \(\text{ PG }(3, q)\), \(q\) even. Discrete Math. 311, 239–243 (2011)Google Scholar
  22. 22.
    Zuanni, F.: A characterization of the external lines of a hyperoval in \(\text{ PG }(3, q)\)\(q\) even. Discrete Math. 312, 1257–1259 (2012)Google Scholar
  23. 23.
    Zuanni, F.: A characterization of the set of the planes of \(\text{ PG }(4, q)\) which meet a non-singular quadric in a conic. Ars Comb. 105, 225–229 (2012)Google Scholar
  24. 24.
    Zuanni, F.: On sets of type \((m, m+q)_{2}\) in \(\text{ PG }(3,q)\), J. Geom.  https://doi.org/10.1007/s00022-017-0401-3

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial and Information Engineering and EconomicsUniversity of L’AquilaL’AquilaItaly

Personalised recommendations