Advertisement

Mathematical Notes

, Volume 105, Issue 3–4, pp 429–438 | Cite as

Vanishing Ideals over Finite Fields

  • A. TochimaniEmail author
  • R. H. VillarrealEmail author
Article
  • 17 Downloads

Abstract

Let \(\mathbb{F}_{q}\) be a finite field, let \(\mathbb{X}\) be a subset of the projective space ℙs−1 over \(\mathbb{F}_{q}\) parametrized by rational functions, and let I(\((\mathbb{X})\)) be the vanishing ideal of \(\mathbb{X}\). The main result of this paper is a formula for I(\((\mathbb{X})\)) that will allow us to compute (i) the algebraic invariants of I(\((\mathbb{X})\)) and (ii) the basic parameters of the corresponding Reed–Muller-type code.

Keywords

vanishing ideal rational parametrization finite field Reed–Muller-type code 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Harris, Algebraic Geometry. A First Course (Springer-Verlag, New York, 1992).zbMATHGoogle Scholar
  2. 2.
    M. González-Sarabia, C. Rentería, and H. Tapia-Recillas, “Reed-Muller-type codes over the Segre variety,” Finite Fields Appl. 8 (4), 511–518 (2002).MathSciNetzbMATHGoogle Scholar
  3. 3.
    T. Sauer, “Polynomial interpolation in several variables: lattices, differences, and ideals,” in Topics in Multivariate Approximation and Interpolation (Stud. Comput. Math., Elsevier B. V., Amsterdam, 2006), Vol. 12, pp. 191–230.Google Scholar
  4. 4.
    D. Grayson and M. Stillman, Macaulay 2, 1996. Available via anonymous ftp from math.uiuc.edu.
  5. 5.
    C. Rentería, A. Simis, and R. H. Villarreal, “Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields,” Finite Fields Appl. 17 (1), 81–104 (2011).MathSciNetzbMATHGoogle Scholar
  6. 6.
    D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms (Springer-Verlag, New York, 1992).zbMATHGoogle Scholar
  7. 7.
    M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2 (Springer-Verlag, Berlin, 2005).zbMATHGoogle Scholar
  8. 8.
    M. Tsfasman, S. Vladut, and D. Nogin, Algebraic Geometric Codes: Basic Notions (American Mathematical Society, Providence, RI, 2007).zbMATHGoogle Scholar
  9. 9.
    H. Matsumura, Commutative Ring Theory (Cambridge University Press, Cambridge, 1986).zbMATHGoogle Scholar
  10. 10.
    G. M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra (Springer, Berlin, 2008).zbMATHGoogle Scholar
  11. 11.
    R. H. Villarreal, Monomial Algebras (CRC Press, Boca Raton, FL, 2015).zbMATHGoogle Scholar
  12. 12.
    N. Jacobson, Basic Algebra I (W. H. Freeman and Company, New York, 1996).zbMATHGoogle Scholar
  13. 13.
    N. Alon, “Combinatorial Nullstellensatz,” Combin. Probab. Comput. 8 (1), 7–29 (1999).MathSciNetzbMATHGoogle Scholar
  14. 14.
    J. Martínez-Bernal, Y. Pitones, and R. H. Villarreal, “Minimum distance functions of graded ideals and Reed–Muller-type codes,” J. Pure Appl. Algebra 221 (2), 251–275 (2017).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Mathematics Department of Center for Research and Advanced StudiesNational Polytechnic InstituteMexico CityMexico

Personalised recommendations