Designs, Codes and Cryptography

, Volume 68, Issue 1–3, pp 25–32 | Cite as

A small minimal blocking set in PG(n, p t ), spanning a (t − 1)-space, is linear

  • Peter Sziklai
  • Geertrui Van de Voorde


In this paper, we show that a small minimal blocking set with exponent e in PG(n, p t ), p prime, spanning a (t/e − 1)-dimensional space, is an \({\mathbb{F}_{p^e}}\) -linear set, provided that p > 5(t/e)−11. As a corollary, we get that all small minimal blocking sets in PG(n, p t ), p prime, p > 5t − 11, spanning a (t − 1)-dimensional space, are \({\mathbb{F}_p}\) -linear, hence confirming the linearity conjecture for blocking sets in this particular case.


Blocking set Linearity conjecture Linear set 

Mathematics Subject Classification (2010)



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  1. 1.
    Ball S.: The number of directions determined by a function over a finite field. J. Comb. Theory Ser. A 104(2), 341–350 (2003)MATHCrossRefGoogle Scholar
  2. 2.
    Blokhuis A.: On the size of a blocking set in PG(2, p). Combinatorica 14(1), 111–114 (1994)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Blokhuis A., Ball S., Brouwer A.E., Storme L., Szőnyi T.: On the number of slopes of the graph of a function defined on a finite field. J. Comb. Theory Ser. A 86(1), 187–196 (1999)MATHCrossRefGoogle Scholar
  4. 4.
    Blokhuis A., Lovász L., Storme L., Szőnyi T.: On multiple blocking sets in Galois planes. Adv. Geom. 7(1), 39–53 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Heim U.: Proper blocking sets in projective spaces. Discret. Math. 174(1–3), 167–176 (1997)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Lavrauw M., Storme L., Vande Voorde G.: On the code generated by the incidence matrix of points and k-spaces in PG(n, q) and its dual. Finite Fields Appl. 14(4), 1020–1038 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Lavrauw M., Vande Voorde G.: On linear sets on a projective line. Des. Codes Cryptogr. 56(2–3), 89–104 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Polverino O.: Small blocking sets in PG(2, p 3). Des. Codes Cryptogr. 20(3), 319–324 (2000)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Storme L., Sziklai P.: Linear pointsets and Rédei type k-blocking sets in PG(n, q). J. Algebraic Comb. 14(3), 221–228 (2001)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Storme L., Weiner Zs.: On 1-blocking sets in PG(n, q), n ≥ 3. Des. Codes Cryptogr. 21(1–3), 235–251 (2000)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Szőnyi T.: Blocking sets in desarguesian affine and projective planes. Finite Fields Appl. 3(3), 187–202 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Szőnyi T., Weiner Zs.: Small blocking sets in higher dimensions. J. Comb. Theory Ser. A 95(1), 88–101 (2001)CrossRefGoogle Scholar
  13. 13.
    Sziklai P.: On small blocking sets and their linearity. J. Comb. Theory Ser. A 115(7), 1167–1182 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Van de Voorde G.: On the linearity of higher-dimensional blocking sets. Electron. J. Comb. 17(1), Research Paper 174, 16 pp (2010).Google Scholar

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© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Eötvös Loránd UniversityBudapestHungary
  2. 2.Vrije Universiteit BrusselBrusselBelgium

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