Skip to main content
SpringerLink
Log in

Reed-Muller Type Codes on the Veronese Variety over Finite Fields

  • Conference paper
Coding Theory, Cryptography and Related Areas

Abstract

The a-invariant, the defining ideal, the dimension and the minimal distance of some Reed-Muller type codes arising from the Veronese variety over a finite field are determined. Some examples are provided to illustrate the main results. These codes are a natural generalization of the projective Reed-Muller codes.

Partially supported by COFAA-IPN and SNI-SEP, México

Partially supported by CONACyT grant No.L0076-E9607 and SNI-SEP, México.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Cox, J. Little, D. O’Shea. Ideals, Varieties, and Algorithms. UTM, Springer Verlag, 1992.

    MATH  Google Scholar 

  2. P. Delsarte, J.M. Goethals, F.J. MacWilliams On generalized Reed-Muller codes and their relatives.Inform. and Control, vol. 16 (1970), pp. 403–422.

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Duursma, C. Renteria and H. Tapia-Recillas. Reed-Muller codes on Complete Intersections. Submitted for publication.

    Google Scholar 

  4. A.V. Geramita, M. Kreuzer and L. Robbiano. Cayley-Bacharach schemes and their canonical modules. Transactions of the AMS, vol. 339, number 1, Sept. (1993), pp. 163–189.

    Article  MathSciNet  MATH  Google Scholar 

  5. J.P. Hansen. Zero-dimensional schemes. Proc. Int. Conf., Ravello, 1992. F. Orrechia and L. Chiantini (eds.), Walter de Gruyter, Berlin, (1994).

    Google Scholar 

  6. G. Lachaud. The parameters of the projective Reed-Muller codes. Discrete Mathematics 81 (1990), pp. 217–221.

    Article  MathSciNet  MATH  Google Scholar 

  7. D.R. Grayson, M. Stillman: Macaulay 2 (1998).

    Google Scholar 

  8. D.J. Mercier, R. Rolland. Polynômes homogènes à plusieurs variables sur un corps fini Fq qui s’annulent sur l’espace projectif Pm (Fg). J. of Pure and Applied Algebra 124 (1998), pp. 227–240

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Renteria, H. Tapia-Recillas. Linear codes associated to the ideal of points in Pd and its canonical module. Communications in Algebra 24 (3), (1996), pp. 10831090.

    Article  MathSciNet  Google Scholar 

  10. C. Renteria, H. Tapia-Recillas. Reed-Muller Codes: An Ideal Theory Approach. Communications in Algebra, 25(2), (1997), pp. 401–413.

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Renteria, H. Tapia-Recillas. The a-invariant of some Reed-Muller Codes. To appear in AAECC journal.

    Google Scholar 

  12. A.B. Sörensen. Projective Reed-Muller codes. IEEE Trans. on Inf. Theory, vol. 37, No. 6, Nov. (1991), pp. 1567–1576.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ESFM, IPN, México

    C. Rentería

  2. Dpto.Mat, UAM-I, México

    H. Tapia-Recillas

Authors
  1. C. Rentería
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Tapia-Recillas
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fachbereich Informatik, Technische Universität Darmstadt, Alexanderstrasse 10, 64283, Darmstadt, Germany

    Johannes Buchmann

  2. Department of Mathematics, Technical University of Denmark, Bldg. 303, 2800, Lyngby, Denmark

    Tom Høholdt

  3. Fachbereich 6, Mathematik und Informatik, Universität Gesamthochschule Essen, 45117, Essen, Germany

    Henning Stichtenoth

  4. Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55-532, C.P. 09340, México, D.F., México

    Horacio Tapia-Recillas

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Rentería, C., Tapia-Recillas, H. (2000). Reed-Muller Type Codes on the Veronese Variety over Finite Fields. In: Buchmann, J., Høholdt, T., Stichtenoth, H., Tapia-Recillas, H. (eds) Coding Theory, Cryptography and Related Areas. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57189-3_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-57189-3_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66248-8

  • Online ISBN: 978-3-642-57189-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation