Kernels and p-Kernels of p r -ary 1-Perfect Codes | Designs, Codes and Cryptography Skip to main content
Springer Nature Link
Log in

Kernels and p-Kernels of pr-ary 1-Perfect Codes

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

Abstract

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over \(\mathbb{F}_{q} = GF(q)\) as well as over the prime field \(\mathbb{F}_{p}\), are established. Q-ary 1-perfect codes of length n=(qm − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.

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

  • S. V. Avgustinovich and F. I. Solov’eva, On projections of perfect binary codes, In Proceedings of the 7th Joint Swedish-Russian International Workshop on Information Theory, St. Petersburg, Russia (1995) pp. 25–26

  • S.V. Avgustinovich F.I. Solov’eva (1996) ArticleTitleOn non-systematic perfect binary codes Problems of Information Transmission. 32 IssueID3 258–261

    Google Scholar 

  • S.V. Avgustinovich F.I. Solov’eva O. Heden (2003) ArticleTitleOn the ranks and kernels problem of perfect codes Problems of Information Transmission. 39 IssueID4 30–34

    Google Scholar 

  • H. Bauer B. Ganter F. Hergert (1983) ArticleTitleAlgebraic techniques for nonlinear codes Combinatorica. 3 21–33

    Google Scholar 

  • I.F. Blake R.C. Mullin (1975) The Mathematical Theory of Coding Academic Press New York

    Google Scholar 

  • G. Cohen I. Honkala S. Litsyn A. Lobstein (1997) Covering Codes North-Holland Amsterdam

    Google Scholar 

  • T. Etzion A. Vardy (1994) ArticleTitlePerfect binary codes: constructions, properties, and enumeration IEEE Transactions on Information Theory. 40 754–763 Occurrence Handle10.1109/18.335887

    Article  Google Scholar 

  • T. Etzion A. Vardy (1998) ArticleTitleOn perfect codes and tilings: problems and solutions SIAM Journal of Discrete Mathematics. 11 IssueID2 205–223 Occurrence Handle10.1137/S0895480196309171

    Article  Google Scholar 

  • T. Etzion (1996) ArticleTitleNonequivalent q-ary perfect codes SIAM Journal of Discrete Mathematics. 9 IssueID3 413–423 Occurrence Handle10.1137/S0895480194277903

    Article  Google Scholar 

  • F.I. MacWilliams N.J. Sloane (1977) The Theory of Error-Correcting Codes North-Holland New York

    Google Scholar 

  • M. LeVan, Designs and Codes, Ph.D. Thesis, Auburn Univesity (1995)

  • B. Lindström (1969) ArticleTitleOn group and nongroup perfect codes in q symbols Mathematical Scandinavian. 25 149–158

    Google Scholar 

  • K.T. Phelps M. Le ParticleVan (1995) ArticleTitleKernels of nonlinear Hamming codes Designs, Codes and Cryptography. 6 247–257

    Google Scholar 

  • K. T. Phelps M. LeVan (1999) ArticleTitleSwitching equivalence classes of perfect codes Designs, Codes and Cryptography 16 179–184

    Google Scholar 

  • K.T. Phelps M. Villanueva (2002) ArticleTitleRanks of q-ary 1-perfect codes Designs, Codes and Cryptography. 27 139–144

    Google Scholar 

  • K.T. Phelps M. Villanueva (2002) ArticleTitleOn perfect codes: rank and kernel Designs, Codes and Cryptography. 27 183–194

    Google Scholar 

  • K. T. Phelps, J. Rifà and M. Villanueva, The switching construction and kernels of q-ary 1-perfect codes, In Proceedings of the ACCT-VIII, September 9–14, St. Petersburg, Russia (2002) pp. 222–225

  • K. T. Phelps, J. Rifà and M. Villanueva, Kernels of q-ary 1-perfect codes, In Proceedings of the Workshop on Coding and Cryptography (WCC), March 24–28, Versailles, France (2003) pp. 375–382

  • G.S. Shapiro D.L. Slotnik (1959) ArticleTitleOn the mathematical theory of error correcting codes IBM Journal of Research and Development. 3 IssueID1 25–34

    Google Scholar 

  • J. Schönheim (1968) ArticleTitleOn linear and nonlinear single-error-correcting q-ary perfect codes Information and Control. 12 23–26 Occurrence Handle10.1016/S0019-9958(68)90167-8

    Article  Google Scholar 

  • J.L. Vasil’ev (1963) ArticleTitleOn nongroup close-packed codes Problemy Kibernetiki. 8 337–339

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Discrete and Statistical Sciences, Auburn University, Auburn, Al, 36849-5307, USA

    K. T. Phelps

  2. Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    J. Rifà

  3. Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    M. Villanueva

Authors
  1. K. T. Phelps
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Rifà
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Villanueva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Villanueva.

Additional information

Communicated by: I.F. Blake

Rights and permissions

Reprints and permissions

About this article

Cite this article

Phelps, K.T., Rifà, J. & Villanueva, M. Kernels and p-Kernels of pr-ary 1-Perfect Codes. Des Codes Crypt 37, 243–261 (2005). https://doi.org/10.1007/s10623-004-3989-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-004-3989-x

Keywords

Search

Navigation