Codes Based on Complete Graphs | Designs, Codes and Cryptography
Skip to main content
Springer Nature Link
Log in

Codes Based on Complete Graphs

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We consider the problem of embedding the even graphicalcode based on the complete graph on n vertices intoa shortening of a Hamming code of length 2m - 1,where m=h(n) should be as small as possible. Asit turns out, this problem is equivalent to the existence problemfor optimal codes with minimum distance 5, and optimal embeddingscan always be realized as graphical codes based on K n.As a consequence, we are able to determine h(n)exactly for all n of the form 2k +1 and to narrow down the possibilities in general to two or threeconceivable values.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North Holland: Amsterdam (1976).

    Google Scholar 

  2. J. G. Bredeson and S. L. Hakimi, Decoding of graph theoretic codes, IEEE Trans. Inform. Th., Vol. 13 (1967) pp. 348–349.

    Google Scholar 

  3. A. E. Brouwer and T. Verhoeff, An updated table of minimum-distance bounds forbinary linear codes, IEEE Trans. Inform. Th., Vol. 39 (1993) pp. 662–680.

    Google Scholar 

  4. S. L. Hakimi and J. G. Bredeson, Graph theoretic error-correcting codes, IEEE Trans. Inform. Th., Vol. 14 (1968) pp. 584–591.

    Google Scholar 

  5. D. Jungnickel and S. A. Vanstone, Graphical Codes Revisited, IEEE Trans. Inform. Th., to appear.

  6. D. Jungnickel and S. A. Vanstone, An application of difference sets to a problem concerning graphical codes, J. Stat. Planning and Inference, to appear.

  7. D. Jungnickel and S. A. Vanstone, Graphical codes: A tutorial, Bull. ICA, to appear.

  8. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland: Amsterdam (1977).

    Google Scholar 

  9. J. H. van Lint, Introduction to Coding Theory, Springer: New York (1982).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl für Angewandte Mathematik II, Universität Augsburg, D-86135, Augsburg, Germany

    Dieter Jungnickel

  2. Dipartimento di Matematica, Universitàdi Roma ``La Sapienza'', 2, Piazzale Aldo Moro, I-00185, Roma, Italy

    Marialuisa J. de Resmini & Scott A. Vanstone

Authors
  1. Dieter Jungnickel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marialuisa J. de Resmini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Scott A. Vanstone
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jungnickel, D., Resmini, M.J.d. & Vanstone, S.A. Codes Based on Complete Graphs. Designs, Codes and Cryptography 8, 159–165 (1996). https://doi.org/10.1023/A:1018089026819

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018089026819

Keywords

Search

Navigation