Ricerche di Matematica

, Volume 63, Issue 2, pp 347–353 | Cite as

Subregular spreads and blocking sets

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Abstract

We prove that any subregular spread in \(PG(3,q)\) has an indicator set which defines a minimal blocking set in \(PG(2,q^2)\) contained in three nonconcurrent lines (i.e., a rank three blocking set) and we discuss some examples.

Keywords

Subregular spreads Indicator sets Translation planes  Redéi blocking sets 

Mathematics Subject Classification (2010)

51E21 51E20 51A40 
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References

  1. 1.
    Bader, L.: Some new examples of flocks of \(Q^+(3, q)\). Geom. Dedic. 27, 213–218 (1988)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bader, L., Lunardon, G.: Desarguesian spreads. Ric. Mat. 60, 15–38 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bader, L., Lunardon, G.: A remark on directions in \(AG(2, q^2)\) (submitted)Google Scholar
  4. 4.
    Ball, S.: The number of directions determined by a function over a finite field. J. Comb. Theory (A) 104, 341–350 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Ball, S., Blokhuis, A., Brouwer, A.E., Storme, L., Szönyi, T.: On the number of slopes of the graph of a function defined on a finite field. J. Comb. Theory (A) 86, 187–196 (1999)CrossRefMATHGoogle Scholar
  6. 6.
    Blokhuis, A., Sziklai, P., Szönyi, T.: Blocking sets in projective spaces. In: De Beule, J., Storme, L. (eds.) Current Research Topics in Galois Geometry, pp. 57–80. Nova Scienze, Commack (2010)Google Scholar
  7. 7.
    Biliotti, M., Jha, V., Johnson, N.L.: Foundations of translation planes, Monogr. Textbooks Pure Appl Math, vol. 243, pp. 542 Marcel Dekker, New York, Basel (2001)Google Scholar
  8. 8.
    Dembowski, P.: Finite Geometries. Springer, Berlin (1968)CrossRefMATHGoogle Scholar
  9. 9.
    Gácz, A., Lovász, L., Szónyi, T.: Directions in \(AG(2, p^2)\). Innov. Incid. Geom. 6–7, 189–201 (2009)Google Scholar
  10. 10.
    Lunardon, G.: Normal spreads. Geom. Dedic. 75, 245–261 (1999)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Lunardon, G.: Blocking sets and semifields. J. Comb. Theory A 113, 1172–1188 (2006)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Harrach, N.V., Mengyán, C.: Minimal blocking sets in \(PG(2, q)\) arising from a generalized construction of Megyesi. Innov. Incid. Geom. 6–7, 211–226 (2007–2008)Google Scholar
  13. 13.
    Orr, W.F.: A characterization of subregular spreads in finite 3-space. Geom. Dedic. 5, 43–50 (1976)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Rédei, L.: Lacunary Polynomials Over Finite Fields, pp 256. North-Holland, Amsterdam (1973)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”Università di Napoli “Federico II”NaplesItaly

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