Skip to main content
SpringerLink
Log in

Refinements of Levenshtein Bounds in q-ary Hamming Spaces

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

Abstract

We develop refinements of the Levenshtein bound in q-ary Hamming spaces by taking into account the discrete nature of the distances versus the continuous behavior of certain parameters used by Levenshtein. We investigate the first relevant cases and present new bounds. In particular, we derive generalizations and q-ary analogs of the MacEliece bound. Furthermore, we provide evidence that our approach is as good as the complete linear programming and discuss how faster are our calculations. Finally, we present a table with parameters of codes which, if exist, would attain our bounds.

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. Delsarte, P. and Levenstein, V.I., Association Schemes and Coding Theory, IEEE Trans. Inform. Theory, 1998, vol. 14, no. 6, pp. 2477–2504.

    Article  MathSciNet  MATH  Google Scholar 

  2. Levenshtein, V.I., Designs as Maximum Codes in Polynomial Metric Spaces, Acta Appl. Math., 1992, vol. 29, no. 1–2, pp. 1–82.

  3. Levenshtein, V.I., Universal Bounds for Codes and Designs, Handbook of Coding Theory, Pless, V.S. and Huffman, W.C., Eds., Amsterdam: Elsevier, 1998, vol. I, ch. 6, pp. 499–648.

    MATH  Google Scholar 

  4. Delsarte, P., An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep. Suppl., 1973, no. 10. Translated under the title Algebraicheskii podkhod k skhemam otnoshenii teorii kodirovaniya, Moscow: Mir, 1976.

    MATH  Google Scholar 

  5. Levenshtein, V.I., Krawtchouk Polynomials and Universal Bounds for Codes and Designs in Hamming Spaces, IEEE Trans. Inform. Theory, 1995, vol. 41, no. 5, pp. 1303–1321.

    Article  MathSciNet  MATH  Google Scholar 

  6. 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.

    MATH  Google Scholar 

  7. Tietäväinen, A., Bounds for Binary Codes Just Outside the Plotkin Range, Inform. Control, 1980, vol. 47, no. 2, pp. 85–93.

    Article  MathSciNet  MATH  Google Scholar 

  8. Krasikov, I. and Litsyn, S., Linear Programming Bounds for Codes of Small Sizes, Europ. J. Combin., 1997, vol. 18, no. 6, pp. 647–656.

    Article  MathSciNet  MATH  Google Scholar 

  9. Sidel’nikov, V.M., On Mutual Correlation of Sequences, Dokl. Akad. Nauk SSSR, 1971, vol. 196, no. 3, pp. 531–534 [Soviet Math. Doklady (Engl. Transl.), 1971, vol. 12, no. 1, pp. 197–201].

    MathSciNet  MATH  Google Scholar 

  10. Perttula, A., Bounds for Binary and Nonbinary Codes Slightly Outside of the Plotkin Range, PhD Thesis, Tampere Univ. of Technology, Tampere, Finland, 1982.

    Google Scholar 

  11. Szegő, G., Orthogonal Polynomials, New York: Amer. Math. Soc., 1939.

    MATH  Google Scholar 

  12. Barg, A. and Nogin, D.Yu., Spectral Approach to Linear Programming Bounds on Codes, Probl. Peredachi Inf., 2006, vol. 42, no. 2, pp. 12–25 [Probl. Inf. Trans. (Engl. Transl.), 2006, vol. 42, no. 2, pp. 77–89].

    MathSciNet  MATH  Google Scholar 

  13. Boyvalenkov, P. and Danev, D., On Linear Programming Bounds for Codes in Polynomial Metric Spaces, Probl. Peredachi Inf., 1998, vol. 34, no. 2, pp. 16–31 [Probl. Inf. Trans. (Engl. Transl.), 1998, vol. 34, no. 2, pp. 108–120].

    MathSciNet  MATH  Google Scholar 

  14. Krasikov, I. and Litsyn, S., On Integral Zeros of Krawtchouk Polynomials, J. Combin. Theory Ser. A, 1996, vol. 74, no. 1, pp. 71–99.

    Article  MathSciNet  MATH  Google Scholar 

  15. Stroeker, R.J. and de Weger, B.M.M., On Integral Zeroes of Binary Krawtchouk Polynomial, Nieuw Arch. Wisk. (4), 1999, vol. 17, no. 2, pp. 175–186.

    MathSciNet  MATH  Google Scholar 

  16. Bounds for Parameters of Codes, Sage Project (Software for Algebra and Geometry Experimentation), https://doi.org/doc.sagemath.org/html/en/reference/coding/sage/coding/code_bounds.html.

  17. Barg, A. and Jaffe, D., Numerical Results on the Asymptotic Rate of Binary Codes, Codes and Association Schemes, Barg, A. and Litsyn, S., Eds., DIMACS Ser., vol. 56, Providence, R.I.: Amer. Math. Soc., 2001, pp. 25–32.

    Book  MATH  Google Scholar 

  18. Samorodnitsky, A., On the Optimum of Delsarte’s Linear Program, J. Combin. Theory Ser. A, 2001, vol. 96, no. 2, pp. 261–287.

    Article  MathSciNet  MATH  Google Scholar 

  19. Gijswijt, D., Schrijver, A., and Tanaka, H., New Upper Bounds for Nonbinary Codes Based on the Terwilliger Algebra and Semidefinite Programming, J. Combin. Theory Ser. A, 2006, vol. 113, no. 8, pp. 1719–1731.

    Article  MathSciNet  MATH  Google Scholar 

  20. Schrijver, A., New Code Upper Bounds from the Terwilliger Algebra and Semidefinite Programming, IEEE Trans. Inform. Theory, 2005, vol. 51, no. 8, pp. 2859–2866.

    Article  MathSciNet  MATH  Google Scholar 

  21. Litjens, B., Polak, S., and Schrijver, A., Semidefinite Bounds for Nonbinary Codes Based on Quadruples, Des. Codes Cryptography, 2017, vol. 84, no. 1–2, pp. 87–100.

    Google Scholar 

  22. Brouwer, A.E., Tables of Codes [electronic]. Available online at https://doi.org/www.win.tue.nl/~aeb/.

  23. Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge: Cambridge Univ. Press, 2004.

    Book  MATH  Google Scholar 

  24. Kerdock, A.M., A Class of Low-Rate Nonlinear Binary Codes, Inform. Control, 1972, vol. 20, no. 2, pp. 182–187.

    Article  MathSciNet  MATH  Google Scholar 

  25. Agrell, E., Vardy, A., and Zeger, K., A Table of Upper Bounds for Binary Codes, IEEE Trans. Inform. Theory, 2001, vol. 47, no. 7, pp. 3004–3006.

    Article  MathSciNet  MATH  Google Scholar 

  26. Best, M.R., Brouwer, A.E., MacWilliams, F.J., Odlyzko, A.M., and Sloane, N.J.A., Bounds for Binary Codes of Length Less than 25, IEEE Trans. Inform. Theory., 1978, vol. 24, no. 1, pp. 81–93.

    Article  MathSciNet  MATH  Google Scholar 

  27. Hedayat, A., Sloane, N.J.A., and Stufken, J., Orthogonal Arrays: Theory and Applications, New York: Springer, 1999.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    P. Boyvalenkov

  2. Faculty of Engineering, South-Western University, Blagoevgrad, Bulgaria

    P. Boyvalenkov

  3. Department of Electrical Engineering and Department of Mathematics, Linköping University, Linköping, Sweden

    D. Danev

  4. Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria

    M. Stoyanova

Authors
  1. P. Boyvalenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Danev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Stoyanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Boyvalenkov.

Additional information

Dedicated to the memory of Prof. V.I. Levenshtein (1935–2017)

Original Russian Text © P. Boyvalenkov, D. Danev, M. Stoyanova, 2018, published in Problemy Peredachi Informatsii, 2018, Vol. 54, No. 4, pp. 35–50.

Supported in part by the Bulgarian NSF, contract DN02/2-13.12.2016.

Supported in part by the Swedish Research Council (VR) and ELLIIT.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boyvalenkov, P., Danev, D. & Stoyanova, M. Refinements of Levenshtein Bounds in q-ary Hamming Spaces. Probl Inf Transm 54, 329–342 (2018). https://doi.org/10.1134/S0032946018040026

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946018040026

Search

Navigation