Skip to main content
SpringerLink
Log in

Bounds on the minimum code distance for nonbinary codes based on bipartite graphs

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

Abstract

The minimum distance of codes on bipartite graphs (BG codes) over GF(q) is studied. A new upper bound on the minimum distance of BG codes is derived. The bound is shown to lie below the Gilbert-Varshamov bound when q ≤ 32. Since the codes based on bipartite expander graphs (BEG codes) are a special case of BG codes and the resulting bound is valid for any BG code, it is also valid for BEG codes. Thus, nonbinary (q ≤ 32) BG codes are worse than the best known linear codes. This is the key result of the work. We also obtain a lower bound on the minimum distance of BG codes with a Reed-Solomon constituent code and a lower bound on the minimum distance of low-density parity-check (LDPC) codes with a Reed-Solomon constituent code. The bound for LDPC codes is very close to the Gilbert-Varshamov bound and lies above the upper bound for BG codes.

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. Tanner, R.M., A Recursive Approach to Low Complexity Codes, IEEE Trans. Inform. Theory, 1981, vol. 27, no. 5, pp. 533–547.

    Article  MathSciNet  MATH  Google Scholar 

  2. Sipser, M. and Spielman, D.A., Expander Codes, IEEE Trans. Inform. Theory, 1996, vol. 42, no. 6, pp. 1710–1722.

    Article  MathSciNet  MATH  Google Scholar 

  3. Lubotzky, A., Phillips, R., and Sarnak, P., Ramanujan Graphs, Combinatorica, 1988, vol. 8, no. 3, pp. 261–277.

    Article  MathSciNet  MATH  Google Scholar 

  4. Margulis, G.A., Explicit Constructions of Concentrators, Probl. Peredachi Inf., 1973, vol. 9, no. 4, pp. 71–80 [Probl. Inf. Trans. (Engl. Transl.), 1973, vol. 9, no. 4, pp. 325–332].

    MathSciNet  MATH  Google Scholar 

  5. Zémor, G., On Expander Codes, IEEE Trans. Inform. Theory, 2001, vol. 47, no. 2, pp. 835–837.

    Article  MathSciNet  MATH  Google Scholar 

  6. Skachek, V. and Roth, R., Generalized Minimum Distance Iterative Decoding of Expander Codes, in Proc. 2003 IEEE Information Theory Workshop, Paris, France, 2003, pp. 245–248.

  7. Barg, A. and Zémor, G., Distance Properties of Expander Codes, IEEE Trans. Inform. Theory, 2006, vol. 52, no. 1, pp. 78–90.

    Article  MathSciNet  Google Scholar 

  8. Skachek, V., Minimum Distance Bounds for Expander Codes, in Information Theory and Applications Workshop (ITA’2008), San Diego, USA, 2008, pp. 366–370.

  9. Boutros, J., Pothier, O., and Zémor, G., Generalized Low Density (Tanner) Codes, in Proc. IEEE Int. Conf. on Communications (ICC), Vancouver, Canada, 1999, vol. 1, pp. 441–445.

  10. Lentmaier, M. and Zigangirov, K., On Generalized Low-Density Parity-Check Codes Based on Hamming Component Codes, IEEE Commun. Lett., 1999, vol. 3, no. 8, pp. 248–250.

    Article  Google Scholar 

  11. Ben-Haim, Y. and Litsyn, S., Upper Bounds on the Rate of LDPC Codes as a Function of Minimum Distance, IEEE Trans. Inform. Theory, 2006, vol. 52, no. 5, pp. 2092–2100.

    Article  MathSciNet  Google Scholar 

  12. Roth, R. and Skachek, V., Improved Nearly-MDS Expander Codes, IEEE Trans. Inform. Theory, 2006, vol. 52, no. 8, pp. 3650–3661.

    Article  MathSciNet  Google Scholar 

  13. Peterson, W.W. and Weldon, E.J., Jr., Error-Correcting Codes, Cambridge: MIT Press, 1972. Translated under the title Kody, ispravlyayushchie oshibki, Moscow: Mir, 1976.

    MATH  Google Scholar 

  14. Bassalygo, L.A., New Upper Bounds for Error Correcting Codes, Probl. Peredachi Inf., 1965, vol. 1, no. 4, pp. 41–44 [Probl. Inf. Trans. (Engl. Transl.), 1965, vol. 1, no. 4, pp. 32–35].

    MathSciNet  MATH  Google Scholar 

  15. McEliece, R.J., Rodemich, E.R., Rumsey, H., Jr., and Welch, L.R., New Upper Bounds on the Rate of a Code via the Delsarte-MacWilliams Inequalities, IEEE Trans. Inform. Theory, 1977, vol. 23, no. 2, pp. 157–166.

    Article  MathSciNet  MATH  Google Scholar 

  16. Gallager, R.G., Low-Density Parity-Check Codes, Cambridge: MIT Press, 1963. Translated under the title Kody s maloi plotnost’yu proverok na chetnost’, Moscow: Mir, 1966.

    Google Scholar 

  17. Barg, A., Mazumdar, A., and Zémor, G., Weight Distribution and Decoding of Codes on Hypergraphs, Adv. Math. Commun., 2008, vol. 2, no. 4, pp. 433–450.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    A. A. Frolov & V. V. Zyablov

Authors
  1. A. A. Frolov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Zyablov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Frolov.

Additional information

Original Russian Text © A.A. Frolov, V.V. Zyablov, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 4, pp. 27–42.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frolov, A.A., Zyablov, V.V. Bounds on the minimum code distance for nonbinary codes based on bipartite graphs. Probl Inf Transm 47, 327–341 (2011). https://doi.org/10.1134/S0032946011040028

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946011040028

Keywords

Search

Navigation