Skip to main content
SpringerLink
Log in

The group of permutation automorphisms of a q-ary hamming code

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

Abstract

We prove that the group of permutation automorphism of a q-ary Hamming code of length n = (q m − 1)/(q − 1) is isomorphic to the unitriangular group UT m (q) if the code has a parity-check matrix composed of all columns of the form (0 ...0 1 * ... *)T. We also show that the group of permutation automorphisms of a cyclic Hamming code cannot be isomorphic to UT m (q). We thus show that equivalent codes can have different permutation automorphism groups.

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. 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 

  2. Huffman, W.C., Codes and Groups, Handbook of Coding Theory, Pless, V.S. and Huffman, W.C., Eds., Amsterdam: Elsevier, 1998, ch. 6, pp. 1345–1440.

    Google Scholar 

  3. Constantinescu, I. and Heise, W., On the Conept of Code-Isomorphy, J. Geom., 1996, vol. 57, no. 1–2, pp. 63–69.

    Article  MATH  MathSciNet  Google Scholar 

  4. Phelps, K.T., Every Finite Group Is the Automorphism Group of Some Perfect Code, J. Combin. Theory, Ser. A, 1986, vol. 43, no. 1, pp. 45–51.

    Article  MATH  MathSciNet  Google Scholar 

  5. Shapiro, G.S. and Slotnik, D.L., On the Mathematical Theory of Error Correcting Codes, IBM J. Res. Dev., 1959, vol. 3, pp. 68–72.

    Article  Google Scholar 

  6. Avgustinovich, S.V. and Solov’eva, F.I., Perfect Binary Codes with Trivial Automorphism Group, in Proc. Int. Workshop on Inf. Theory, Killarney, Ireland, 1998, pp. 114–115.

  7. Malyugin, S.A., Perfect Codes with Trivial Automorphism Group, Proc. 2nd Int. Workshop on Optimal Codes and Related Topics, Sozopol, Bulgaria, 1998, Sofia, 1998, pp. 163–167.

  8. Vasil’ev, Yu.L., On Nongroup Closely Packed Codes, Probl. Kibern., 1962, vol. 8, pp. 337–339.

    Google Scholar 

  9. Avgustinovich, S.V., Solov’eva, F.I., and Heden, O., On the Structure of Symmetry Groups of Vasil’ev Codes, Probl. Peredachi Inf., 2005, vol. 41, no. 2, pp. 42–49 [Probl. Inf. Trans. (Engl. Transl.), 2005, vol. 41, no. 2, pp. 105–112].

    MathSciNet  Google Scholar 

  10. Solov’eva, F.I. and Topalova, S.T., On Automorphism Groups of Perfect Binary Codes and Steiner Triple Systems, Probl. Peredachi Inf., 2000, vol. 36, no. 4, pp. 53–58 [Probl. Inf. Trans. (Engl. Transl.), 2000, vol. 36, no. 4, pp. 331–335].

    MathSciNet  Google Scholar 

  11. Solov’eva, F.I. and Topalova, S.T., Perfect Binary Codes and Steiner Triple Systems with Maximal Orders of Automorphism Groups, Diskretn. Anal. Issled. Oper., Ser. 1, 2000, vol. 7, no. 4, pp. 101–110.

    MATH  MathSciNet  Google Scholar 

  12. Malyugin, S.A., On the Order of Automorphism Group of Perfect Binary Codes, Diskretn. Anal. Issled. Oper., Ser. 1, 2000, vol. 7, no. 4, pp. 91–100.

    MATH  MathSciNet  Google Scholar 

  13. Lindström, B., On Group and Nongroup Perfect Codes in q Symbols, Math. Scand., 1969, vol. 25, pp. 149–158.

    MATH  Google Scholar 

  14. Markov, A.A., On Transformations without Error Propagation, Izbrannye trudy (SelectedWorks), vol. II: Teoriya algorifmov i konstruktivnaya matematika. Matematicheskaya logika. Informatika i smezhnye voprosy (Theory of Algorithms and Constructive Mathematics. Mathematical Logic. Information Science and Related Topics), Nagornyi, N.M., Ed., Moscow: MCCME, 2003, pp. 70–93.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novosibirsk State University, Novosibirsk, Russia

    E. V. Gorkunov

Authors
  1. E. V. Gorkunov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. V. Gorkunov.

Additional information

Original Russian Text © E.V. Gorkunov, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 4, pp. 18–25.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gorkunov, E.V. The group of permutation automorphisms of a q-ary hamming code. Probl Inf Transm 45, 309–316 (2009). https://doi.org/10.1134/S0032946009040024

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946009040024

Keywords

Search

Navigation