Bounding the Minimum Distance of Affine Variety Codes Using Symbolic Computations of Footprints

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10495)

Abstract

We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in [12, Example 3.2]. Among the codes that we construct almost all have parameters as good as the best known codes according to [9] and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound [7, 10] from Gröbner basis theory and for this purpose we develop a new method where we inspired by Buchbergers algorithm perform a series of symbolic computations.

Keywords

Affine variety codes Buchberger’s algorithm Klein curve Minimum distance 

Notes

Acknowledgments

The authors gratefully acknowledge the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367). They are also grateful to Department of Mathematical Sciences, Aalborg University for supporting a one-month visiting professor position for the second listed author. The research of Ferruh Özbudak has been funded by METU Coordinatorship of Scientific Research Projects via grant for projects BAP-01-01-2016-008 and BAP-07-05-2017-007.

References

  1. 1.
    Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, vol. 4. Springer, New York (2015)CrossRefMATHGoogle Scholar
  2. 2.
    Feng, G.L., Rao, T.R.N.: Decoding algebraic-geometric codes up to the designed minimum distance. IEEE Trans. Inform. Theory 39(1), 37–45 (1993)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Feng, G.L., Rao, T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inform. Theory 40(4), 1003–1012 (1994)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Feng, G.L., Rao, T.R.N.: Improved geometric Goppa codes part I: basic theory. IEEE Trans. Inform. Theory 41(6), 1678–1693 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fitzgerald, J., Lax, R.F.: Decoding affine variety codes using Gröbner bases. Des. Codes Crypt. 13(2), 147–158 (1998)CrossRefMATHGoogle Scholar
  6. 6.
    Geil, O.: Evaluation codes from an affine variety code perspective. In: Martínez-Moro, E., Munuera, C., Ruano, D. (eds.) Advances in Algebraic Geometry Codes. Coding Theory and Cryptology, vol. 5, pp. 153–180. World Scientific, Singapore (2008)CrossRefGoogle Scholar
  7. 7.
    Geil, O., Høholdt, T.: Footprints or generalized Bezout’s theorem. IEEE Trans. Inform. Theory 46(2), 635–641 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Geil, O., Martin, S.: An improvement of the Feng-Rao bound for primary codes. Des. Codes Crypt. 76(1), 49–79 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes (2007). http://www.codetables.de. Accessed 20 Apr 2017
  10. 10.
    Høholdt, T.: On (or in) Dick Blahut‘s’ footprint’. Codes, Curves and Signals, pp. 3–9 (1998)Google Scholar
  11. 11.
    Høholdt, T., van Lint, J.H., Pellikaan, R.: Algebraic geometry codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998)Google Scholar
  12. 12.
    Kolluru, M.S., Feng, G.L., Rao, T.R.N.: Construction of improved geometric Goppa codes from Klein curves and Klein-like curves. Appl. Algebra Engrg. Comm. Comput. 10(6), 433–464 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Salazar, G., Dunn, D., Graham, S.B.: An improvement of the Feng-Rao bound on minimum distance. Finite Fields Appl. 12, 313–335 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark
  2. 2.Department of Mathematics and Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey

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