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Extended Norm-Trace Codes with Optimized Correction Capability

  • Maria Bras-Amorós
  • Michael E. O’Sullivan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4851)

Abstract

We consider a generalization of the codes defined by norm and trace functions on finite fields introduced by Olav Geil. The codes in the new family still satisfy Geil’s duality properties stated for norm-trace codes. That is, it is easy to find a minimal set of parity checks guaranteeing correction of a given number of errors, as well as the set of monomials generating the corresponding code. Furthermore, we describe a way to find the minimal set of parity checks and the corresponding generating monomials guaranteeing correction at least of generic errors. This gives codes with even larger dimensions.

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References

  1. 1.
    Bras-Amorós, M., O’Sullivan, M.E.: Duality for Some Families of Correction Capability Optimized Evaluation Codes (2007)Google Scholar
  2. 2.
    Geil, O.: On Codes From Norm-Trace Curves. Finite Fields Appl. 9(3), 351–371 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Koetter, R.: On the Determination of Error Values for Codes From a Class of Maximal Curves. In: Proc. 35-th Allerton Conference on Communication, Control, and Computing, pp. 44–53 (1997)Google Scholar
  4. 4.
    Lee, K., O’Sullivan, M.E.: List Decoding of Hermitian Codes Using Groebner Bases (2006)Google Scholar
  5. 5.
    Hoeholdt, T., van Lint, J.H., Pellikaan, R.: Algebraic Geometry Codes. In: Handbook of Coding Theory, vol. I, pp. 871–961. North-Holland, Amsterdam (1998)Google Scholar
  6. 6.
    O’Sullivan, M.E.: New Codes for the Berlekamp-Massey-Sakata Algorithm. Finite Fields Appl. 7(2), 293–317 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Geil, O., Pellikaan, R.: On the Structure of Order Domains. Finite Fields Appl. 8(3), 369–396 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Geil, O.: Codes Based on an F<Subscript>q</Subscript>-Algebra. PhD thesis, Aalborg University (1999)Google Scholar
  9. 9.
    Little, J.B.: The Ubiquity of Order Domains for the Construction Of Error Control Codes. Adv. Math. Commun. 1(1), 151–171 (2007)Google Scholar
  10. 10.
    Sakata, S.: Extension of Berlekamp-Massey Algorithm to n Dimensions. IEEE Trans. Inform. Theory 34(5), 1332–1340 (1988)CrossRefMathSciNetGoogle Scholar
  11. 11.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Duursma, I.M.: Majority Coset Decoding. IEEE Trans. Inform. Theory 39(3), 1067–1070 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bras-Amorós, M., O’Sullivan, M.E.: The Correction Capability of the Berlekamp-Massey-Sakata Algorithm With Majority Voting. Appl. Algebra Engrg. Comm. Comput. 17(5), 315–335 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Geil, O., Hoeholdt, T.: Footprints or Generalized Bezout’s Theorem. IEEE Trans. Inform. Theory 46(2), 635–641 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications, 1st edn. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  16. 16.
    Bourbaki, N.: Commutative Algebra, ch. 1–7. Elements of Mathematics. Springer, Berlin (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Maria Bras-Amorós
    • 1
  • Michael E. O’Sullivan
    • 2
  1. 1.Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili 
  2. 2.Department of Mathematics and Statistics, San Diego State University 

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