Skip to main content
SpringerLink
Log in

On the graph of a function in many variables over a finite field

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

Abstract

Some improved bounds on the number of directions not determined by a point set in the affine space AG(k, q) are presented. More precisely, if there are more than p e(q − 1) directions not determined by a set of q k-1 points \({\mathcal S}\) then every hyperplane meets \({\mathcal S}\) in 0 modulo p e+1 points. This bound is shown to be tight in the case p e = q s and when q = p es sets of q k-1 points that do not meet every hyperplane in 0 modulo p e+1 points and have a little less than p e(q − 1) non-determined directions are constructed.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

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 Combin Theory Ser A 104: 341–350 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ball S, Govaerts P and Storme L. On ovoids of parabolic quadrics. Des Codes Cryptogr 38: 131–145 (2006)

    Article  MathSciNet  Google Scholar 

  3. Ball S and Lavrauw M. How to use Rédei polynomials in higher dimensional spaces. Le Matematiche (Catania) 59: 39–52 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Ball S and Lavrauw M. On the graph of a function in two variables over a finite field. J Algebraic Combin 23: 243–253 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blokhuis A, Ball S, Brouwer AE, Storme L and Szőnyi T. On the number of slopes of the graph of a function defined over a finite field. J Combin Theory Ser A 86: 187–196 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Storme L and Sziklai P. Linear point sets and Rédei type k-blocking sets in PG(n, q). J Algebraic Combin 14: 221–228 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. De Matematica Aplicada IV, Univ. Politecnica De Catalunya, Modul C3, Campus Nord, C/Jordi Girona, 1, Barcelona, E-08034, Spain

    Simeon Ball

Authors
  1. Simeon Ball
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Simeon Ball.

Additional information

The author acknowledges the support of the Ramon y Cajal programme and the project MTM2005-08990-C02-01 of the Spanish Ministry of Science and Education and the project 2005SGR00256 of the Catalan Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ball, S. On the graph of a function in many variables over a finite field. Des. Codes Cryptogr. 47, 159–164 (2008). https://doi.org/10.1007/s10623-007-9108-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-007-9108-z

Keywords

AMS Classifications

Search

Navigation