Advertisement

Journal of Algebraic Combinatorics

, Volume 42, Issue 3, pp 725–744 | Cite as

On translation spreads of \(H(q)\)

  • Giuseppe Marino
  • Olga Polverino
Article

Abstract

Using the connection between translation spreads of \(H(q)\) and linear sets (see Cardinali et al. in Eur J Comb 23:367–376, 2002; Lunardon and Polverino in J Algebraic Comb 18:255–262, 2003), some results on the existence of translation spreads of \(H(q)\) are given, improving classification results contained in Offer (Spreads and ovoids of the split Cayley hexagon, 2000) and in Bonoli and Polverino (Discrete Math 296:129–142, 2005).

Keywords

Generalized hexagon Spread Twisted cubic Linear set 

Notes

Acknowledgments

This work was supported by the Research Project of MIUR (Italian Office for University and Research) “Strutture Geometriche, Combinatoria e loro Applicazioni”.

References

  1. 1.
    Bader, L., Lunardon, G.: Generalized hexagons and BLT-sets. In: Buekenhout, Beutelspacher, De Clerck, Doyen, Hirschfeld, Thas, (eds.) Finite Geometry and Combinatorics, pp. 5–16. Cambridge University Press, Cambridge (1994)Google Scholar
  2. 2.
    Ball, S., Blokhuis, A., Lavrauw, M.: On the classification of semifield flocks. Adv. Math. 180, 104–111 (2003)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bloemen, I., Thas, J.A., Van Maldeghem, H.: Translation ovoids of generalized quadrangles and hexagons. Geom. Dedicata 72, 19–62 (1998)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Blokhuis, A., Lavrauw, M.: Scattered spaces with respect to a spread in \(PG(n, q)\). Geom. Dedicata 81(1–3), 231–243 (2000)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bonoli, G., Polverino, O.: The twisted cubic of \(PG(3, q)\) and translation spreads of \(H(q)\). Discrete Math. 296, 129–142 (2005)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Cannon, J., Playoust, C.: An Introduction to MAGMA. University of Sydney Press, Sydney (1993)Google Scholar
  7. 7.
    Cardinali, I., Lunardon, G., Polverino, O., Trombetti, R.: Translation spreads of the classical generalized hexagon. Eur. J. Comb. 23, 367–376 (2002)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Carlitz, L.: A theorem on “ordered” polynomials in a finite field. Acta Arithmetica VII, 167–172 (1962)Google Scholar
  9. 9.
    De Smet, V., Van Maldeghem, H.: The finite Moufang hexagons coordinatized. Beiträge Algebra Geom. 34, 217–232 (1993)MATHGoogle Scholar
  10. 10.
    Hirschfeld, J.W.P.: Finite Projective Spaces of Three Dimensions. Oxford University Press, Oxford (1985)MATHGoogle Scholar
  11. 11.
    Hirschfeld, J.W.P., Korchmáros, G., Torres, F.: Algebraic Curves Over a Finite Field. Princeton University Press, Princeton (2013)Google Scholar
  12. 12.
    Kantor, W.M.: Ovoids and translation planes. Can. J. Math. 34, 1195–1207 (1982)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Lavrauw, M.: Scattered Spaces with Respect to Spreads, and Eggs in Finite Projective Spaces. Dissertation, Eindhoven University of Technology, Eindhoven (2001)Google Scholar
  14. 14.
    Lavrauw, M.: Sublines of prime order contained in the set of internal points of a conic. Des. Codes Cryptogr. 38(1), 113–123 (2006)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Lavrauw, M., Marino, G., Polverino, O., Trombetti, R.: \(\mathbb{F}_q\)-pseudoreguli of \(PG(3, q^3)\) and scattered semifields of order \(q^6\). Finite Fields Appl. 17, 225–239 (2011)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Lavrauw, M., Marino, G., Polverino, O., Trombetti, R.: Solution to an isotopism question concerning rank 2 semifields. J. Comb. Des. 23, 60–77 (2015)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Lavrauw, M., Van de Voorde, G.: On linear sets on a projective line. Des. Codes Cryptogr. 56, 89–104 (2010)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Lavrauw, M., Van de Voorde, G.: Scattered linear sets and pseudoreguli. Electron. J. Comb. 20(1) (2013)Google Scholar
  19. 19.
    Lunardon, G.: Flocks, ovoids of \(Q(4, q)\) and designs. Geom. Dedicata 66, 163–173 (1997)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Lunardon, G.: Translation ovoids. J. Geom. 76(1–2), 200–215 (2003)MATHMathSciNetGoogle Scholar
  21. 21.
    Lunardon, G., Marino, G., Polverino, O., Trombetti, R.: Maximum scattered linear sets of pseudoregulus type and the Segre variety \({\cal S}_{n, n}\). J. Algebraic Comb 39, 807–831 (2014)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lunardon, G., Polverino, O.: On the twisted cubic of \(PG(3, q)\). J. Algebraic Comb. 18, 255–262 (2003)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lunardon, G., Polverino, O.: Translation ovoids of orthogonal polar spaces. Forum Math. 16, 663–669 (2004)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Luyckx, D., Thas, J.A.: Flocks and locally hermitian \(1-\)systems of \(Q(6, q)\). Finite Geometries, Proceedings of the Fourth Isle of Thorns Conference, Brighton, UK, April 2000. Dordrecht: Kluwer Academic Publishers. Dev. Math. 3, 257–275 (2001)Google Scholar
  25. 25.
    Marino, G., Polverino, O., Trombetti, R.: On \({\mathbb{F}}_q\)-linear sets of PG\((3, q^3)\) and semifields. J. Comb. Theory Ser. A 114, 769–788 (2007)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Offer, A.D.: Spreads and Ovoids of the Split Cayley Hexagon. Ph.D. Thesis, The University of Adelaide, Adelaide (2000)Google Scholar
  27. 27.
    Offer, A.D.: Translation ovoids and spreads of the generalized hexagon \(H(q)\). Geom. Dedicata 85(1–3), 135–145 (2001)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Offer, A.D.: Translation spreads of the split Cayley hexagon. Adv. Geom. 3(2), 105–121 (2003)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Polverino, O.: Linear sets in finite projective spaces. Discrete Math. 310, 3096–3107 (2010)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Thas, J.A.: Polar spaces, generalized hexagons and perfect codes. J. Comb. Theory Ser. A 29, 87–93 (1980)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Thas, J.A.: Generalized quadrangles and flocks of cones. Eur. J. Comb. 8, 441–452 (1987)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Thas, J.A.: Generalized quadrangles of order \((s, s^2)\). II. J. Comb. Theory Ser. A 79, 223–254 (1997)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Tits, J.: Sur la trialité et certains groupes qui sen déduisent. Publ. Mat. de l’Inst des hautes études scientifiques 2, 37–60 (1959)Google Scholar
  34. 34.
    Van Maldeghem, H.: Generalized Polygons. Birkhäuser Verlag, Basel (1998)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaSeconda Università degli Studi di NapoliCasertaItaly

Personalised recommendations