Advertisement

Eurocode ’92 pp 231-253 | Cite as

On the Efficient Decoding of Algebraic-Geometric Codes

  • R. Pellikaan
Part of the International Centre for Mechanical Sciences book series (CISM, volume 339)

Abstract

This talk is intended to give a survey on the existing literature on the decoding of algebraic-geometric codes. Although the motivation originally was to find an efficient decoding algorithm for algebraic-geometric codes, the latest results give algorithms which can be explained purely in terms of linear algebra. We will treat the following subjects:
  1. 1.

    The decoding problem

     
  2. 2.

    Decoding by error location

     
  3. 3.

    Decoding by error location of algebraic-geometric codes

     
  4. 4.

    Majority coset decoding

     
  5. 5.

    Decoding algebraic-geometric codes by solving the key equation

     
  6. 6.

    Improvements of the complexity

     

Keywords

Linear Code Cyclic Code Hermitian Curve Goppa Code Principal Divisor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Arimoto, Encoding and decoding of p-ary group codes and the correction system, Information Processing in Japan 2 (1961), 320–325. in Japanese.MathSciNetGoogle Scholar
  2. [2]
    E.R. Berlekamp, Algebraic coding theory, McGraw-Hill, New York 1968.MATHGoogle Scholar
  3. [3]
    E.R. Berlekamp, R.J. McEliece and H.C.A. van Tilborg, On the inherent intractibility of certain coding problems, IEEE Trans. Inform. Theory 24 (1978), 384–386.CrossRefMATHGoogle Scholar
  4. [4]
    P. Bours, J.C.M. Janssen, M. van Asperdt and H.C.A. van Tilborg, Algebraic decoding beyond eBCH of some binary cyclic codes, when e eBCH, IEEE Trans. Inform. Theory 36 (1990), 214–222.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    J. Bruck and M. Naor, The hardness of decoding linear codes with preprocessing, IEEE Trans. Inform. Theory 36 (1990), 381–385.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, PhD Thesis, University of Innsbruck, Austria, 1965.Google Scholar
  7. [7]
    Ph. Carbonne and A. Thiong Ly, Minimal exponent for Pellikaan’s decoding algorithm, Proceedings Eurocode 92.Google Scholar
  8. [8]
    C. Chevalley, Introduction to the theory of algebraic functions in one variable, Math. Surveys VI, Providence, AMS 1951.Google Scholar
  9. [9]
    I.M. Duursma, Algebraic decoding using special divisors, to appear in IEEE Trans. Inform. Theory.Google Scholar
  10. [10]
    I.M. Duursma, Majority coset decoding, to appear in IEEE Trans. Inform. Theory.Google Scholar
  11. [11]
    I.M. Duursma, On the decoding procedure of Feng and Rao, Proceedings ACCT-3, Voneshta Voda, June 1992.Google Scholar
  12. [12]
    I.M. Duursma and R. Kötter, On error locating pairs for cyclic codes, preprint October 1992.Google Scholar
  13. [13]
    D. Ehrhard, Über das Dekodieren Algebraisch-Geometrischer Codes, PhD Thesis, University of Düsseldorf, July 1991.MATHGoogle Scholar
  14. [14]
    D. Ehrhard, Decoding algebraic-geometric codes by solving a key equation, in the Proceedings AGCT-3, H. Stichtenoth and M.A. Tsfasman (eds.), Luminy 1991, Springer Lect. Notes. 1518 (1992), 18–25.Google Scholar
  15. [15]
    D. Ehrhard, Achieving the designed error capacity in decoding algebraic-geometric codes, to appear in IEEE Trans. Inform. Theory.Google Scholar
  16. [16]
    G.-L. Feng and T.R.N. Rao, Decoding of algebraic geometric codes up to the designed minimum distance, to appear in IEEE Trans. Inform. Theory.Google Scholar
  17. [17]
    G.-L. Feng and T.R.N. Rao, A novel approach for construction of algebraic-geometric codes from affine plane curves, University of Southwestern Louisiana, preprint 1992.Google Scholar
  18. G.-L. Feng and K.K. Tzeng, A generalization of the Berlekamp-Massey algorithm for multisequence shift register synthesis with application to decoding cyclic codes, IEEE Trans. Inform. Theory 37 (1991) 1274–1287.Google Scholar
  19. [19]
    G.-L. Feng and K.K. Tzeng, Decoding cyclic and BCH codes up to the actual minimum distance using nonrecurrent syndrome dependence relations, IEEE Trans. Inform. Theory 37 (1991), 1716–1723.CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    V.D. Goppa, Codes associated with divisors, Probl. Peredachi Inform. 13(1) (1977), 33–39. Translation: Probl. Inform. Transmission 13 (1977), 22–26.MathSciNetGoogle Scholar
  21. [21]
    V.D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR 259 (1981), 1289–1290, Translation: Soviet Math. Dokl. 24 (1981), 170–172.MATHGoogle Scholar
  22. [22]
    V.D. Goppa, Algebraico-geometric codes, Izv. Akad. Nauk SSSR 46 (1982), Translation: Math. USSR Izvestija 21 (1983), 75–91.MATHGoogle Scholar
  23. [23]
    V.D. Goppa, Codes and information, Russian Math. Surveys 39 (1984), 87–141.CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    V.D. Goppa, Geometry and codes, Mathematics and its Applications 24, Kluwer Acad. Publ., Dordrecht, 1991.Google Scholar
  25. [25]
    C. Dahl Jensen, Codes and geometry, PhD Thesis, Technical University of Denmark, May 1991Google Scholar
  26. [26]
    J. Justesen, K.J. Larsen, H.Elbrond Jensen, A. H.vemose and T. Hoholdt, Construction and decoding of a class of algebraic geometric codes, IEEE Trans. Inform. Theory 35 (1989), 811–821.MATHMathSciNetGoogle Scholar
  27. [27]
    J. Justesen, K.J. Larsen, H.Elbrond Jensen and T. HOholdt, Fast decoding of codes from algebraic plane curves, IEEE Trans. Inform. Theory 38 (1992), 111–119.CrossRefMATHMathSciNetGoogle Scholar
  28. [28]
    J. Justesen, H.Elbrond Jensen and T. Hoholdt, On the number of correctable errors for some AG-codes, to appear in IEEE Trans. Inform. Theory.Google Scholar
  29. [29]
    R. Kötter, A unified description of an error locating procedure for linear codes, Proceedings ACCT-3, Voneshta Voda, June 1992.Google Scholar
  30. [30]
    V. Yu. Krachkovskii, Decoding of codes on algebraic curves, Odessa, preprint 1988 in Russian.Google Scholar
  31. [31]
    D. Le Brigand, Decoding of codes on hyperelliptic curves, Proceedings Eurocode 90, G.D. Cohen and P. Charpin (eds.), Lect. Notes in Comp. Sc. 514 (1991) 126–134.Google Scholar
  32. [32]
    J.H. van Lint, Algebraic geometric codes, in Coding Theory and Design Theory, part I, D. Ray-Chaudhuri (ed.), IMA Volumes Math. Appl. 21 Springer-Verlag, Berlin etc. (1990)Google Scholar
  33. [33]
    J.H. van Lint and G. van der Geer, Introduction to coding theory and algebraic geometry, DMV Seminar 12, Birkhäuser Verlag, Basel Boston Berlin, 1988.Google Scholar
  34. [34]
    J.H. van Lint and T.A. Springer, Generalized Reed-Solomon codes from algebraic geometry, IEEE Trans. Inform. Theory 33 (1987), 305–309.CrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    J.H. van Lint and R.M. Wilson, On the minimum distance of cyclic codes, IEEE Trans. Inform. Theory 32 (1996), 23–40.CrossRefGoogle Scholar
  36. [36]
    J.L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory 15 (1969), 122–127.CrossRefMATHGoogle Scholar
  37. [37]
    F.J. McWilliams and N.J.A. Sloane, The theory of error-correcting codes, North-Holland Math. Library 16, North-Holland, Amsterdam, 1977.Google Scholar
  38. [38]
    C. Moreno, Algebraic curves over finite fields, Cambridge Tracts in Math. 97, Cambridge Un. Press, 1991.Google Scholar
  39. [39]
    R. Pellikaan, On decoding linear codes by error correcting pairs, preprint Eindhoven University of Technology, 1988.Google Scholar
  40. [40]
    R. Pellikaan, On a decoding algorithm for codes on maximal curves, IEEE Trans. Inform. Theory 35 (1989), 1228–1232.CrossRefMATHMathSciNetGoogle Scholar
  41. [41]
    R. Pellikaan, On the decoding by error location and the number of dependent error positions, Discrete Math. 106/107 (1992), 369–381.Google Scholar
  42. [42]
    R. Pellikaan, B.-Z. Shen and G.J.M. van Wee, Which linear codes are algebraic-geometric ?, IEEE Trans. Inform. Theory. IT-37 (1991), 583–602.Google Scholar
  43. [43]
    W.W. Peterson, Encoding and error-correction procedures for the Bose-Chauduri codes, IEEE Trans. Inform. Theory 6 (1960), 459–470.CrossRefGoogle Scholar
  44. [44]
    S.C. Porter, Decoding codes arising from Goppa’s construction on algebraic curves, Thesis, Yale University, dec. 1988.Google Scholar
  45. [45]
    S.C. Porter, B.-Z. Shen and R. Pellikaan, On decoding geometric Goppa codes using an extra place, IEEE Trans. Inform. Theory. IT-38 (1992), 1663–1676.Google Scholar
  46. [46]
    D. Rotillon and J.A. Thiong Ly, Decoding codes on the Klein quartic, Proceedings Eurocode 90, G.D. Cohen and P. Charpin (eds.), Lect. Notes in Comp. Sc. 514 (1991), 135–150.Google Scholar
  47. [47]
    S. Sakata, On determining the independent point set for doubly periodic arrays and encoding two-dimensional cyclic codes and their duals, IEEE Trans. Inform. Theory 27 (1981), 556–565.CrossRefMATHMathSciNetGoogle Scholar
  48. [48]
    S. Sakata, Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array, Journal of Symbolic Computation, 5 (1988), 321–337.CrossRefMATHMathSciNetGoogle Scholar
  49. [49]
    S. Sakata, Extension of the Berlekamp-Massey algorithm to N dimensions, Information and Computation, 84 (1990), 207–239.CrossRefMATHMathSciNetGoogle Scholar
  50. [50]
    S. Sakata, Decoding binary 2-D cyclic codes by the 2-D Berlekamp-Massey algorithm, IEEE Trans. Inform. Theory 37 (1991), 1200–1203.CrossRefMATHMathSciNetGoogle Scholar
  51. [51]
    B.-Z. Shen, Solving a congruence on a graded algebra by a subresultant sequence and its application, to appear in Journ. of Symbolic Computation.Google Scholar
  52. [52]
    B.-Z. Shen, On encoding and decoding of the codes from Hermitian curves, to appear in Cryptography and Coding III, the IMA Conference Proceedings Series, M. Ganley (ed.), Oxford University Press.Google Scholar
  53. [53]
    B.-Z. Shen, Constructing syndromes for the codes from Hemitian curves and a decoding approach, preprint Eindhoven University of Technology, 1992.Google Scholar
  54. [54]
    B.-Z. Shen, Algebraic-geometric codes and their decoding algorithm, PhD Thesis, Eindhoven University of Technology, September 1992.MATHGoogle Scholar
  55. [55]
    A.N. Skorobogatov and S.G. Vlâdutt, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory 36 (1990), 1051–1060.CrossRefMATHMathSciNetGoogle Scholar
  56. [56]
    H. Stichtenoth, A note on Hermitian codes over GF(ga), IEEE Trans. Inform. Theory 34 (1988), 1345–1348.CrossRefMathSciNetGoogle Scholar
  57. [57]
    H. Stichtenoth, Algebraic function fields and codes, to appear in Universitext, Springer-Verlag, 1993.Google Scholar
  58. [58]
    H.J. Tiersma, Codes comming from Hermitian curves, IEEE Trans. Inform. Theory 33 (1987), 605–609.CrossRefMATHMathSciNetGoogle Scholar
  59. [59]
    M.A. Tsfasman and S.G. Vl.dut, Algebraic-geometric codes, Mathematics and its Applications 58, Kluwer Acad. Publ., Dordrecht, 1991.Google Scholar
  60. [60]
    M.A. Tsfasman, S.G. Vlâdut and T. Zink, Modular curves, Shimura curves and Goppa codes, better than Varshamov-Gilbert bound, Math. Nachrichten 109 (1982), 21–28.CrossRefMATHGoogle Scholar
  61. [61]
    S.G. Vl.dut, On the decoding of algebraic-geometric codes over GF(q) for q 16, IEEE Trans. Inform. Theory 36 (1990), 1461–1463.CrossRefMathSciNetGoogle Scholar
  62. [62]
    T. Yaghoobian and I.F. Blake, Hermitian codes as generalized Reed-Solomon codes, Designs, Codes and Cryptogrphy 2 (1992), 15–18.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • R. Pellikaan
    • 1
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations