Skip to main content
SpringerLink
Log in

Minimum distance of relative Reed–Muller codes

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

We describe a construction of error-correcting codes on a fibration over a curve defined over a finite field, which may be considered as a relative version of the classical Reed–Muller code. In the case of complete intersections in a projective bundle, we give an explicit lower bound for the minimum distance.

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. Arason J.K., Elman R., Jacob B.: On indecomposable vector bundles. Commun. Alg. 20, 1323–1351 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aubry, Y.: Reed–Muller codes associated to projective varieties. In: Lecture Notes Mathematics (Luminy, France, June 17–21 1991), vol. 1518, pp. 4–17. Springer, Berlin (1992)

  3. Aubry, Y., Perret, M.: A Weil theorem for singular curves. In: Arithmetic, Geometry and Coding Theory (Luminy, 1993), pp. 1–17. de Gruyter, Berlin (1993)

  4. Ghorpade S.R., Lachaud G.: Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields. Moscow Math. J. 2, 589–631 (2002)

    MATH  MathSciNet  Google Scholar 

  5. Hansen S.H.: Error-correcting codes from higher-dimensional varieties. Finite Fields Appl. 7, 530–552 (2001)

    Article  MATH  Google Scholar 

  6. Lachaud, G.: Number of points of plane sections and linear codes defined on algebraic varieties. In: Arithmetic, Geometry and Coding Theory (Luminy, 1993), pp.77–104. de Gruyter, Berlin (1993)

  7. Manin Y.I., Vladut S.G.: Linear codes and modular curves. J.Soviet Math. 30, 2611–2643 (1985)

    Article  MATH  Google Scholar 

  8. Miyaoka Y.: The Chern classes and Kodaira dimension of a minimal variety. Adv. Stud. Pure Math. 10, 449–476 (1987)

    MathSciNet  Google Scholar 

  9. Nakashima T.: Codes on Grassmann bundles and related varieties. J. Pure Appl. Alg. 199, 235–244 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Nakashima T.: Error-correcting codes on projective bundles. Finite Fields Appl. 12, 222–231 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tsfasman M.A., Vladut S.G.: Algebraic-Geometric Codes. Kluwer, Dordrecht (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women’s University, Mejirodai, Bunkyoku, Tokyo, 112-8681, Japan

    Tohru Nakashima

Authors
  1. Tohru Nakashima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tohru Nakashima.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nakashima, T. Minimum distance of relative Reed–Muller codes. AAECC 20, 123–132 (2009). https://doi.org/10.1007/s00200-009-0093-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-009-0093-5

Keywords

Search

Navigation