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)


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.


Affine variety codes Buchberger’s algorithm Klein curve Minimum distance 



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.


  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). 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

Personalised recommendations