A small minimal blocking set in PG(n, p t ), spanning a (t − 1)-space, is linear | Designs, Codes and Cryptography Skip to main content
Springer Nature Link
Log in

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

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MATH  Google Scholar 

  2. Blokhuis A.: On the size of a blocking set in PG(2, p). Combinatorica 14(1), 111–114 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  4. Blokhuis A., Lovász L., Storme L., Szőnyi T.: On multiple blocking sets in Galois planes. Adv. Geom. 7(1), 39–53 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Heim U.: Proper blocking sets in projective spaces. Discret. Math. 174(1–3), 167–176 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lavrauw M., Vande Voorde G.: On linear sets on a projective line. Des. Codes Cryptogr. 56(2–3), 89–104 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Polverino O.: Small blocking sets in PG(2, p 3). Des. Codes Cryptogr. 20(3), 319–324 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Storme L., Weiner Zs.: On 1-blocking sets in PG(n, q), n ≥ 3. Des. Codes Cryptogr. 21(1–3), 235–251 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Szőnyi T.: Blocking sets in desarguesian affine and projective planes. Finite Fields Appl. 3(3), 187–202 (1997)

    Article  MathSciNet  Google Scholar 

  12. Szőnyi T., Weiner Zs.: Small blocking sets in higher dimensions. J. Comb. Theory Ser. A 95(1), 88–101 (2001)

    Article  Google Scholar 

  13. Sziklai P.: On small blocking sets and their linearity. J. Comb. Theory Ser. A 115(7), 1167–1182 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117, Budapest, Hungary

    Peter Sziklai

  2. Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussel, Belgium

    Geertrui Van de Voorde

Authors
  1. Peter Sziklai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Geertrui Van de Voorde
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Geertrui Van de Voorde.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sziklai, P., Van de Voorde, G. A small minimal blocking set in PG(n, p t), spanning a (t − 1)-space, is linear. Des. Codes Cryptogr. 68, 25–32 (2013). https://doi.org/10.1007/s10623-012-9751-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-012-9751-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation