Skip to main content
SpringerLink
Log in

Distance regularity of Kerdock codes

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

A code is called distance regular, if for every two codewords x, y and integers i, j the number of codewords z such that d(x, z) = i and d(y, z) = j, with d the Hamming distance, does not depend on the choice of x, y and depends only on d(x, y) and i, j. Using some properties of the discrete Fourier transform we give a new combinatorial proof of the distance regularity of an arbitrary Kerdock code. We also calculate the parameters of the distance regularity of a Kerdock code.

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., An Algebraic Approach to the Association Schemes of Coding Theory, Philips Research Reports Supplements 10, Historical Jrl., Ann Arbor (1973).

    Google Scholar 

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

    MATH  Google Scholar 

  3. Brouwer A. E., Cohen A. M., and Neumaier A., Distance-Regular Graphs, Springer-Verlag, Berlin (1989).

    MATH  Google Scholar 

  4. Bannai E. and Ito T., Algebraic Combinatorics. I: Association Schemes, Benjamin, London (1984).

    MATH  Google Scholar 

  5. Levenshtein V. I., “Universal bounds for codes and designs,” in: Handbook of Coding Theory. Vol. 1, Elsevier, Amsterdam, 1998, pp. 499–648.

    Google Scholar 

  6. Topalova S. T., “Distance regularity of some linear codes,” in: Abstracts of the Annual Workshop on Algebraic and Combinatorial Coding Theory, St. Zagora, Bulgaria, 2000, p. 18.

  7. Avgustinovich S. V. and Solov’eva F. I., “On distance regularity of perfect binary codes,” Problems Inform. Transmission, 34, No. 3, 247–249 (1998).

    MATH  MathSciNet  Google Scholar 

  8. Avgustinovich S. V. and Solov’eva F. I., “New constructions and properties of perfect codes,” in: Proc. Intern. Workshop “Discrete Analysis and Operation Research” [in Russian], Novosibirsk, 2000, pp. 5–10.

  9. Solov’eva F. I. and Tokareva N. N., “On distance nonregularity of Preparata codes,” Siberian Math. J., 48, No. 2, 327–333 (2007).

    Article  MathSciNet  Google Scholar 

  10. Sidel’nikov V. M., “Extremal polynomials used in bounds of code volume,” Problems Inform. Transmission, 16, No. 3, 174–186 (1981).

    Google Scholar 

  11. Kerdock A. M., “A class of low-rate non-linear binary codes,” Inform. Control, 20, No. 2, 182–187 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  12. Kantor W. M., “An exponential number of generalized Kerdock codes,” Inform. Control, 53, No. 1–2, 74–80 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  13. Nechaev A. A., “The Kerdock code in cyclic form,” Diskret. Mat., 1, No. 4, 123–139 (1989).

    MATH  MathSciNet  Google Scholar 

  14. Hammons A. R., Kumar P. V., Calderbank A. R., Sloane N. J. A., and Solé P., “The ℤ4-linearity of Kerdock, Preparata, Goethals, and related codes,” IEEE Trans. Inform. Theory, 40, No. 2, 301–319 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  15. Wan Zhe-Xian, Quaternary Codes, World Scientific, Singapore (1997).

    MATH  Google Scholar 

  16. Nordstrom A. W. and Robinson J. P., “An optimum nonlinear code,” Inform. Control, 11, No. 5–6, 613–616 (1967).

    Article  MATH  Google Scholar 

  17. Kantor W. M., “Codes, quadratic forms and finite geometries,” Proc. Sympos. Appl. Math., 50, 153–177 (1995).

    MathSciNet  Google Scholar 

  18. Calderbank A. R., Cameron P. J., Kantor W. M., and Seidel J. J., “ℤ4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets,” Proc. London Math. Soc., 75, 436–480 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  19. Kantor W. M. and Williams M. E., “Symplectic semifield planes and ℤ4-linear codes,” Trans. Amer. Math. Soc., 356, No. 3, 895–938 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  20. Vasil’eva A. Yu., “Strong distance invariance of perfect binary codes,” Diskret. Anal. Issled. Oper. Ser. 1, 9, No. 4, 33–40 (2002).

    MATH  MathSciNet  Google Scholar 

  21. Solov’eva F. I. and Tokareva N. N., “On the property of distance regularity of Kerdock and Preparata codes,” in: Proc. Tenth Intern. Workshop “Algebraic and Combinatorial Coding Theory,” 3–9 September 2006, Zvenigorod, Russia; IITP, Moscow, 2006, pp. 248–251.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    F. I. Solov’eva & N. N. Tokareva

Authors
  1. F. I. Solov’eva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. N. Tokareva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. I. Solov’eva.

Additional information

Original Russian Text Copyright © 2008 Solov’eva F. I. and Tokareva N. N.

The first author was partially supported by the Royal Swedish Academy of Sciences. The second author was supported by the Russian Science Support Foundation, the Integration Project of the Siberian Branch of the Russian Academy of Sciences “A Tree-Like Catalog of Internet Mathematical Resources” (Grant No. 35), and the Russian Foundation for Basic Research (Grants 07-01-00248 and 08-01-00671). Both authors were partially supported by Novosibirsk State University.

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 3, pp. 669–682, May–June, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Solov’eva, F.I., Tokareva, N.N. Distance regularity of Kerdock codes. Sib Math J 49, 539–548 (2008). https://doi.org/10.1007/s11202-008-0051-7

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-008-0051-7

Keywords

Search

Navigation