Skip to main content
SpringerLink
Log in

Decoding of repeated-root cyclic codes up to new bounds on their minimum distance

  • Coding Theory
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

The well-known approach of Bose, Ray-Chaudhuri, and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum Hamming distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bose, R.C. and Ray-Chaudhuri, D.K. On a Class of Error Correcting Binary Group Codes, Inform. Control, 1960, vol. 3, no. 1, pp. 68–79.

    Article  MATH  MathSciNet  Google Scholar 

  2. Hocquenghem, A. Codes correcteurs d’erreurs, Chiffres, 1959, vol. 2, pp. 147–156.

    MATH  MathSciNet  Google Scholar 

  3. Hartmann, C.R.P. Decoding beyond the BCH Bound, IEEE Trans. Inform. Theory, 1972, vol. 18, no. 3, pp. 441–444.

    Article  MATH  Google Scholar 

  4. Hartmann, C.R.P. and Tzeng, K.K. Generalizations of the BCH Bound, Inform. Control, 1972, vol. 20, no. 5, pp. 489–498.

    Article  MATH  MathSciNet  Google Scholar 

  5. Berman, S.D. Semisimple Cyclic and Abelian Codes, Cybernetics, 1967, vol. 3, no. 3, pp. 17–23.

    Article  MathSciNet  Google Scholar 

  6. Massey, J.L. Costello, D.J., and Justesen, J., PolynomialWeights and Code Constructions, IEEE Trans. Inform. Theory, 1973, vol. 19, no. 1, pp. 101–110.

    Article  MATH  MathSciNet  Google Scholar 

  7. Massey, J.L. von Seemann, N., and Schöller, P., Hasse Derivatives and Repeated-Root Cyclic Codes, in Proc. 1986 IEEE Int. Sympos. on Information Theory (ISIT’1986), Ann Arbor, MI,USA, Oct. 6–9, 1986, p. 39.

    Google Scholar 

  8. da Rocha, V.C. On Repeated-Single-Root Constacyclic Codes, Communications and Cryptography: Two Sides of One Tapestry, Blahut, R.E., Costello, D.J., Jr., Maurer U., and Mittelholzer, T., Eds., Boston: Kluwer, 1994, pp. 93–99.

    Chapter  Google Scholar 

  9. Castagnoli, G. On the Minimum Distance of Long Cyclic Codes and Cyclic Redundancy Check Codes, PhD Thesis, Zürich, Switzerland: ETH, 1989.

    Google Scholar 

  10. Castagnoli, G. On the Asymptotic Badness of Cyclic Codes with Block-Lengths Composed from a Fixed Set of Prime Factors, Proc. 6th Int. Conf. on Applied Algebra, Algebraic Algorithms and Error- Correcting Codes (AAECC-6), Rome, Italy, July 4–8, 1988. Mora, T. Ed., Lect. Notes Comput. Sci., vol. 357, Berlin: Springer, 1989, pp. 164–168.

    MathSciNet  Google Scholar 

  11. Castagnoli, G. Massey, J.L., Schöller, P.A., and von Seemann, N., On Repeated-Root Cyclic Codes, IEEE Trans. Inform. Theory, 1991, vol. 37, no. 2, pp. 337–342.

    Article  MATH  MathSciNet  Google Scholar 

  12. Latypov, R.Kh. Checking Matrix of a Cyclic Code Generated by Multiple Roots, Issled. Prikl. Mat., 1979, no. 7, pp. 111–116 [J. Soviet Math. (Engl. Transl.), 1988, vol. 43, no. 3, pp. 2492–2495].

    MATH  MathSciNet  Google Scholar 

  13. van Lint, J. Repeated-Root Cyclic Codes, IEEE Trans. Inform. Theory, 1991, vol. 37, no. 2, pp. 343–345.

    Article  MATH  MathSciNet  Google Scholar 

  14. Zimmermann, K.-H. On Generalizations of Repeated-Root Cyclic Codes, IEEE Trans. Inform. Theory, 1996, vol. 42, no. 2, pp. 641–649.

    Article  MATH  MathSciNet  Google Scholar 

  15. Nedeloaia, C.-S. Weight Distributions of Cyclic Self-Dual Codes, IEEE Trans. Inform. Theory, 2003, vol. 49, no. 6, pp. 1582–1591.

    Article  MATH  MathSciNet  Google Scholar 

  16. Ling, S. Niederreiter, H., and Solé, P., On the Algebraic Structure of Quasi-cyclic Codes IV: Repeated Roots, Des. Codes Cryptogr., 2006, vol. 38, no. 3, pp. 337–361.

    Article  MATH  MathSciNet  Google Scholar 

  17. Dinh, H.Q. Repeated-Root Constacyclic Codes of Length 2ps, Finite Fields Appl., 2012, vol. 18, no. 1, pp. 133–143.

    Article  MATH  MathSciNet  Google Scholar 

  18. Dinh, H.Q. Structure of Repeated-Root Constacyclic Codes of Length 3ps and Their Duals, Discrete Math., 2013, vol. 313, no. 9, pp. 983–991.

    Article  MATH  MathSciNet  Google Scholar 

  19. Zeh, A. and Bezzateev, S.V. A New Bound on the Minimum Distance of Cyclic Codes Using Small- Minimum-Distance Cyclic Codes, Des. Codes Cryptogr., 2014, vol. 71, no. 2, pp. 229–246.

    Article  MATH  MathSciNet  Google Scholar 

  20. Zeh, A. Wachter-Zeh, A., and Bezzateev, S.V., Decoding Cyclic Codes up to a New Bound on the Minimum Distance, IEEE Trans. Inform. Theory, 2012, vol. 58, no. 6, pp. 3951–3960.

    Article  MathSciNet  Google Scholar 

  21. Burton, H. and Weldon, E.J. Cyclic Product Codes, IEEE Trans. Inform. Theory, 1965, vol. 11, no. 3, pp. 433–439.

    Article  MATH  MathSciNet  Google Scholar 

  22. Lin, S. and Weldon, E.J. Further Results on Cyclic Product Codes, IEEE Trans. Inform. Theory, 1970, vol. 16, no. 4, pp. 452–459.

    Article  MATH  MathSciNet  Google Scholar 

  23. Feng, G.-L. and Tzeng, K.K. A Generalized Euclidean Algorithm for Multisequence Shift-Register Synthesis, IEEE Trans. Inform. Theory, 1989, vol. 35, no. 3, pp. 584–594.

    Article  MATH  MathSciNet  Google Scholar 

  24. Hasse, H., Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit vollkommenem Konstantenkörper bei beliebiger Charakteristik, J. Reine Angew. Math., 1936, vol. 175, pp. 50–54.

    MathSciNet  Google Scholar 

  25. Krachkovsky, V.Yu. and Lee, Y.X. Decoding for Iterative Reed–Solomon Coding Schemes, IEEE Trans. Magn., 1997, vol. 33, no. 5, pp. 2740–2742.

    Article  Google Scholar 

  26. Sidorenko, V.R. Schmidt, G., and Bossert, M., Decoding Punctured Reed–Solomon Codes up to the Singleton Bound, in Proc. 7th Int. ITG Conference on Source and Channel Coding (SCC’08), Ulm, Germany, Jan. 14–16, 2008.

    Google Scholar 

  27. Guruswami, V. Linear-Algebraic List Decoding of Folded Reed–Solomon Codes, in Proc. 2011 IEEE 26th Annual Conf. on Computational Complexity (CCC’11), San Jose, CA,USA, June 8–11, 2011, pp. 77–85.

    Google Scholar 

  28. Feng, G.-L. and Tzeng, K.K. A Generalization of the Berlekamp–Massey Algorithm for Multisequence Shift-Register Synthesis with Applications to Decoding Cyclic Codes, IEEE Trans. Inform. Theory, 1991, vol. 37, no. 5, pp. 1274–1287.

    Article  MATH  MathSciNet  Google Scholar 

  29. Zeh, A. and Wachter, A. Fast Multi-Sequence Shift-Register Synthesis with the Euclidean Algorithm, Adv. Math. Commun., 2011, vol. 5, no. 4, pp. 667–680.

    Article  MATH  MathSciNet  Google Scholar 

  30. MacWilliams, F.J. and Sloane, N.J.A. The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.

    Google Scholar 

  31. Zeh, A. Wachter-Zeh, A., Gadouleau, M., and Bezzateev, S.V., Generalizing Bounds on the Minimum Distance of Cyclic Codes Using Cyclic Product Codes, in Proc. 2013 IEEE Int. Sympos. on Information Theory (ISIT’2013), Istanbul, Turkey, July 7–12, 2013, pp. 126–130.

    Chapter  Google Scholar 

  32. Krachkovsky, V.Yu. and Lee, Y.X. Decoding of Parallel Reed–Solomon Codes with Applications to Product and Concatenated Codes, in Proc. 1988 IEEE Int Sympos. on Information Theory (ISIT’98), Cambridge, MA,USA, Aug. 16–21, 1998. pp. 55.

    Google Scholar 

  33. Schmidt, G. Sidorenko, V.R., and Bossert, M., Collaborative Decoding of Interleaved Reed–Solomon Codes and Concatenated Code Designs, IEEE Trans. Inform. Theory, 2009, vol. 55, no. 7, pp. 2991–3012.

    Article  MathSciNet  Google Scholar 

  34. Gopalan, P. Guruswami, V., and Raghavendra, P., List Decoding Tensor Products and Interleaved Codes, SIAM J. Comput., 2011, vol. 40, no. 5, pp. 1432–1462.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Technion, Haifa, Israel

    A. Zeh

  2. Institute of Communications and Navigation, German Aerospace Center (DLR), Berlin, Germany

    M. Ulmschneider

Authors
  1. A. Zeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Ulmschneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Zeh.

Additional information

Original Russian Text © A. Zeh, M. Ulmschneider, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 3, pp. 15–30.

Supported by the German Research Council (DFG) under grants Bo867/22-1 and Ze1016/1-1; the work was initiated when both authors were affiliated with the Institute of Communications Engineering, University of Ulm, Ulm, Germany.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zeh, A., Ulmschneider, M. Decoding of repeated-root cyclic codes up to new bounds on their minimum distance. Probl Inf Transm 51, 217–230 (2015). https://doi.org/10.1134/S0032946015030023

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946015030023

Keywords

Search

Navigation