Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 1161–1174 | Cite as

On intriguing sets of finite symplectic spaces



Some constructions of intriguing sets of finite symplectic spaces are provided. In particular an affirmative answer to an existence question about small tight sets posed in De Beule et al. (Des Codes Cryptogr 50(2):187–201, 2009) is given.


Symplectic polar space Tight set m-ovoids 

Mathematics Subject Classification

Primary 51A50 Secondary 51E20 05B25 51E12 


  1. 1.
    Bamberg J., Kelly S., Law M., Penttila T.: Tight sets and \(m\)-ovoids of finite polar spaces. J. Combin. Theory Ser. A 114(7), 1293–1314 (2007).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bray J., Holt D., Roney-Dougal C.: The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. London Mathematical Society, LNS 407, Cambridge University Press, New York (2013).CrossRefMATHGoogle Scholar
  3. 3.
    Bruen A.A., Hirschfeld J.W.P.: Intersections in projective space. I. Combinatorics. Math. Z. 193(2), 215–225 (1986).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Butler D.K.: On the intersection of ovoids sharing a polarity. Geom. Dedicata 135, 157–165 (2008).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cossidente A.: On Kestenband–Ebert partitions. J. Combin. Des. 5(5), 367–375 (1997).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cossidente A., Culbert C., Ebert G.L., Marino G.: On \(m\)-ovoids of \({\cal{W}}(3, q)\). Finite Fields Appl. 14(1), 76–84 (2008).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cossidente A., Pavese F.: On the geometry of unitary involutions. Finite Fields Appl. 36, 14–28 (2015).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cossidente A., Pavese F.: Subspace codes in \({\rm PG}(2n-1,q)\). Combinatorica. doi: 10.1007/s00493-016-3354-5.
  9. 9.
    Cossidente A., Pavese F.: Intriguing sets of \({\cal{W}}(5, q)\), \(q\) even. J. Combin. Theory Ser. A 127, 303–313 (2014).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    De Beule J., Govaerts P., Hallez A., Storme L.: Tight sets, weighted \(m\)-covers, weighted \(m\)-ovoids, and minihypers. Des. Codes Cryptogr. 50(2), 187–201 (2009).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dye R.H.: Spreads and classes of maximal subgroups of \({\rm GL}_n(q),{\rm SL}_n(q),{\rm PGL}_n(q)\) and \({\rm PSL}_n(q)\). Ann. Mat. Pura Appl. 158(4), 33–50 (1991).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dye R.H.: Partitions and their stabilizers for line complexes and quadrics. Ann. Mat. Pura Appl. (V) 114, 173–194 (1977).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Drudge K.: Proper \(2\)-covers of \({\rm PG}(3, q)\), \(q\) even. Geom. Dedicata 80(1–3), 59–64 (2000).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ebert G.L.: Partitioning projective geometries into caps. Can. J. Math. 37(6), 1163–1175 (1985).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hirschfeld J.W.P.: Projective Geometries over Finite Fields. Oxford Mathematical MonographsOxford Science Publications, The Clarendon Press, Oxford University Press, New York (1998).MATHGoogle Scholar
  16. 16.
    Kelly S.: Constructions of intriguing sets of polar spaces from field reduction and derivation. Des. Codes Cryptogr. 43(1), 1–8 (2007).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kestenband B.C.: Projective geometries that are disjoint unions of caps. Can. J. Math. 32(6), 1299–1305 (1980).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kleidman P., Liebeck M.: The Subgroup Structure of the Finite Classical Groups, vol. 129. London Mathematical Society Lecture Note SeriesCambridge University Press, Cambridge (1990).CrossRefMATHGoogle Scholar
  19. 19.
    Nakić A., Storme L.: Tight sets in finite classical polar spaces. Adv. Geom. 17(1), 109–129 (2017).MathSciNetGoogle Scholar
  20. 20.
    O’Keefe C.M.: Ovoids in \({\rm PG}(3, q)\): a survey. Discret. Math. 151, 171–188 (1996).MathSciNetGoogle Scholar
  21. 21.
    Pavese F.: Geometric constructions of two-character sets. Discret. Math. 338(3), 202–208 (2015).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Segre B.: Forme e geometrie hermitiane, con particolare riguardo al caso finito. Ann. Mat. Pura Appl. 70(4), 1–201 (1965).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Segre B.: On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two. Acta Arith. 5, 315–332 (1959).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Segre B.: Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane. Ann. Mat. Pura Appl. 64, 1–76 (1964).MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica, Informatica ed EconomiaUniversità degli Studi della BasilicataPotenzaItaly
  2. 2.Dipartimento di Meccanica, Matematica e ManagementPolitecnico di BariBariItaly

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