Skip to main content
SpringerLink
Log in

The Order Bound for Toric Codes

  • Conference paper
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5527))

Abstract

In this paper we investigate the minimum distance of generalized toric codes using an order bound like approach. We apply this technique to a family of codes that includes the Joyner code. For some codes in this family we are able to determine the exact minimum distance.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. Beelen, P.: The order bound for general algebraic geometric codes. Finite Fields Appl. 13(3), 665–680 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beelen, P., Pellikaan, R.: The Newton polygon of plane curves with many rational points. Des. Codes Cryptogr. 21(1-3), 41–67 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Danilov, V.I.: The geometry of toric varieties. Russian Math. Surverys 33(2), 97–154 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Duursma, I.M.: Majority coset decoding. IEEE Trans. Inform. Theory 39(3), 1067–1070 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Feng, G.L., Rao, T.R.N.: Improved geometric Goppa codes. I. Basic theory. IEEE Trans. Inform. Theory 41(6, part 1), 1678–1693 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fulton, W.: Introduction to toric varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  7. Hansen, J.P.: Toric surfaces and error-correcting codes. In: Buchmann, J., Høholdt, T., Stichtenoth, H., Tapia-Recillas, H. (eds.) Coding theory, cryptography and related areas (Guanajuato, 1998), pp. 132–142. Springer, Berlin (2000)

    Chapter  Google Scholar 

  8. Hansen, J.P.: Toric varieties Hirzebruch surfaces and error-correcting codes. Appl. Algebra Engrg. Comm. Comput. 13(4), 289–300 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Høholdt, T., van Lint, J.H., Pellikaan, R.: Algebraic geometry of codes. In: Handbook of coding theory, vol. I, II, pp. 871–961. North-Holland, Amsterdam (1998)

    Google Scholar 

  10. Husemöller, D.: Elliptic curves, 2nd edn. Graduate Texts in Mathematics, vol. 111. Springer, New York (2004)

    MATH  Google Scholar 

  11. Joyner, D.: Toric codes over finite fields. Appl. Algebra Engrg. Comm. Comput. 15(1), 63–79 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Little, J., Schenck, H.: Toric surface codes and Minkowski sums. SIAM J. Discrete Math. 20(4), 999–1014 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Little, J., Schwarz, R.: On toric codes and multivariate Vandermonde matrices. Appl. Algebra Engrg. Comm. Comput. 18(4), 349–367 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Martínez-Moro, E., Ruano, D.: Toric codes. In: Advances in Algebraic Geometry Codes. Series on Coding Theory and Cryptology, vol. 5, pp. 295–322. World Scientific, Singapore (2008)

    Chapter  Google Scholar 

  15. Ruano, D.: On the parameters of r-dimensional toric codes. Finite Fields Appl. 13(4), 962–976 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ruano, D.: On the structure of generalized toric codes. J. Symb. Comput. 44(5), 499–506 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Soprunov, I., Soprunova, E.: Toric surface codes and minkowski length of polygons. arXiv:0802.2088v1 [math. AG] (2008)

    Google Scholar 

  18. Serre, J.-P.: Sur le nombre des points rationnels d’une courbe algébrique sur un corps fini. C. R. Acad. Sci. Paris Sér. I Math. 296(9), 397–402 (1983)

    MathSciNet  MATH  Google Scholar 

  19. Tsfasman, M.A., Vlăduţ, S.G.: Algebraic-geometric codes, Dordrecht. Mathematics and its Applications (Soviet Series), vol. 58 (1991)

    Google Scholar 

  20. Waterhouse, W.C.: Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. 2(4), 521–560 (1969)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DTU-Mathematics, Technical University of Denmark, Matematiktorvet, Building 303, 2800 Kgs. Lyngby, Denmark

    Peter Beelen & Diego Ruano

Authors
  1. Peter Beelen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diego Ruano
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avinguda Països Catalans, 26, 43007, Tarragona, Spain

    Maria Bras-Amorós

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

    Tom Høholdt

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Beelen, P., Ruano, D. (2009). The Order Bound for Toric Codes. In: Bras-Amorós, M., Høholdt, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2009. Lecture Notes in Computer Science, vol 5527. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02181-7_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02181-7_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02180-0

  • Online ISBN: 978-3-642-02181-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation