Skip to main content
SpringerLink
Log in

Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes

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

Abstract

We study cardinalities of components of perfect codes and colorings, correlation immune functions, and bent function (sets of ones of these functions). Based on results of Kasami and Tokura, we show that for any of these combinatorial objects the component cardinality in the interval from 2k to 2k+1 can only take values of the form 2k+1 − 2p, where p ∈ {0, ..., k} and 2k is the minimum component cardinality for a combinatorial object with the same parameters. For bent functions, we prove existence of components of any cardinality in this spectrum. For perfect colorings with certain parameters and for correlation immune functions, we find components of some of the above-given cardinalities.

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. Fon-Der-Flaass, D.G., A Bound on Correlation Immunity, Sib. Elektron. Mat. Izv., 2007, vol. 4, pp. 133–135.

    MathSciNet  MATH  Google Scholar 

  2. Tarannikov, Yu.V., On Correlation-Immune and Stable Boolean Functions, Mat. Vopr. Kibern., vol. 11, Moscow: Fizmatlit, 2002, pp. 91–148.

    Google Scholar 

  3. Etzion, T. and Vardy, A., On Perfect Codes and Tilings: Problems and Solutions, SIAM J. Discrete Math., 1998, vol. 11, no. 2, pp. 205–223.

    Article  MathSciNet  MATH  Google Scholar 

  4. Avgustinovich, S.V., Heden, O, and Solov’eva, F.I., On Intersections of Perfect Binary Codes, Bayreuth. Math. Schr. 2005, no. 74, pp. 1–6.

  5. Avgustinovich, S.V., Heden, O., and Solov’eva, F.I., On Intersection Problem for Perfect Binary Codes, Des. Codes Cryptogr., 2006, vol. 39, no. 3, pp. 317–322.

    Article  MathSciNet  MATH  Google Scholar 

  6. Avgustinovich, S.V., Lobstein, A.C., and Solov’eva, F.I., Intersection Matrices for Partitions by Binary Perfect Codes, IEEE Trans. Inform. Theory, 2001, vol. 47, no. 4, pp. 1621–1624.

    Article  MathSciNet  MATH  Google Scholar 

  7. Vasil’ev, Yu.L., Avgustinovich, S.V., and Krotov, D.S., On Shifting Sets in the Binary Hypercube, Diskretn. Anal. Issled. Oper., 2008, vol. 15, no. 3, pp. 11–21 [J. Appl. Ind. Math. (Engl. Transl.), 2009, vol. 3, no. 2, pp. 290–296].

    MathSciNet  MATH  Google Scholar 

  8. Heden, O., Soloveva, F.I., and Mogilnykh, I.Yu., Intersections of Perfect Binary Codes, in Proc. 2010 IEEE Region 8 Int. Conf. on Computational Technologies in Electrical and Electronics Engineering (SIBIRCON), Irkutsk Listvyanka, Russia, 2010. Piscataway: IEEE, 2010, pp. 52–54.

    Chapter  Google Scholar 

  9. Soloveva, F.I. and Los’, A.V., Intersections of q-ary Perfect Codes, Sibirsk. Mat. Zh., 2008, vol. 49, no. 2, pp. 464–474 [Sib. Math. J. (Engl. Transl.), 2008, vol. 49, no. 2, pp. 375–382].

    MathSciNet  Google Scholar 

  10. Kolomeec, N.A. and Pavlov, A.V., Properties of Bent Functions with Minimal Distance, Prikl. Diskr. Mat., 2009, no. 4, pp. 5–20.

  11. Kasami, T. and Tokura, N., On the Weight Structure of Reed-Muller Codes, IEEE Trans. Inform. Theory, 1970, vol. 16, no. 6, pp. 752–759.

    Article  MathSciNet  MATH  Google Scholar 

  12. Logachev, O.A., Sal’nikov, A.A., and Yashchenko, V.V., Bulevy funktsii v teorii kodirovaniya i kriptologii (Boolean Functions in Coding Theory and Cryptology), Moscow: MCCME, 2004.

    MATH  Google Scholar 

  13. 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 

  14. Kasami, T., Tokura, N., and Azumi, S., On the Weight Enumeration of Weights Less than 2.5d of Reed-Muller Codes, Inform. and Control, 1976, vol. 30, no. 4, pp. 380–395.

    Article  MathSciNet  MATH  Google Scholar 

  15. Avgustinovich, S.V., private communication (talk given at the Coding Theory seminar, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences), 2003.

  16. Potapov, V.N., On Perfect Colorings of Boolean n-Cube and Correlation Immune Functions with Small Density, Sib. Elektron. Mat. Izv., 2010, vol. 7, pp. 372–382.

    MathSciNet  Google Scholar 

  17. Avgustinovich, S.V., Heden, O., and Solov’eva, F.I., The Classification of Some Perfect Codes, Des. Codes Cryptogr., 2004, vol. 31, no. 3, pp. 313–318.

    Article  MathSciNet  MATH  Google Scholar 

  18. Phelps, K.T., A General Product Construction for Error Correcting Codes, SIAM J. Algebr. Discrete Methods, 1984, vol. 5, no. 2, pp. 224–228.

    Article  MathSciNet  Google Scholar 

  19. Krotov, D.S. and Potapov, V.N., On Switching Equivalence of n-ary Quasigroups of Order 4 and Perfect Binary Codes, Probl. Peredachi Inf., 2010, vol. 46, no. 3, pp. 22–28 [Probl. Inf. Trans. (Engl. Transl.), 2010, vol. 46, no. 3, pp. 219–224].

    MathSciNet  Google Scholar 

  20. Potapov, V.N., Latin Bitrade, ArXiv e-print arXiv:1104.1295v1, 2011.

  21. Krotov, D.S. and Potapov, V.N., On Multifold MDS and Perfect Codes That Are Not Splittable into Onefold Codes, Probl. Peredachi Inf., 2004, vol. 40, no. 1, pp. 6–14 [Probl. Inf. Trans. (Engl. Transl.), 2004, vol. 40, no. 1, pp. 5–12].

    MathSciNet  Google Scholar 

  22. Fon-Der-Flaas, D.G., Perfect 2-Colorings of a Hypercube, Sibirsk. Mat. Zh., 2007, vol. 48, no. 4, pp. 923–930 [cm[Sib. Math. J. (Engl. Transl.), 2007, vol. 48, no. 4, pp. 740–745]].

    MathSciNet  Google Scholar 

  23. Sarkar, P., Spectral Domain Analysis of Correlation Immune and Resilient Boolean Functions, Cryptology ePrint Archive: Report 2000/049, 2000. Available at http://eprint.iacr.org/2000/049.

  24. Avgustinovich, S.V. and Vasil’eva, A.Yu., Computation of a Centered Function from Its Values on the Middle Layers of the Boolean Cube, Diskretn. Anal. Issled. Oper., Ser. 1, 2003, vol. 10, no. 2, pp. 3–16.

    MathSciNet  MATH  Google Scholar 

  25. Tokareva, N.N., Bent Functions: Results and Applications. A Survey, Prikl. Diskr. Mat., 2009, no. 1, pp. 15–37.

  26. Carlet, C., Boolean Functions for Cryptography and Error Correcting Codes, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Crama, Y. and Hammer, P.L., Eds., Cambridge: Cambridge Univ. Press, 2010, ch. 8, pp. 257–397.

    Google Scholar 

  27. Carlet, C., Two New Classes of Bent Functions, Advances in Cryptology-EUROCRYPT’93. Proc. Workshop on the Theory and Application of Cryptographic Techniques, Lofthus, Norway, Helleseth, T., Ed., Lect. Notes Comp. Sci., vol. 765, Berlin: Springer, 1994, pp. 77–101.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia

    V. N. Potapov

  2. Novosibirsk State University, Novosibirsk, Russia

    V. N. Potapov

Authors
  1. V. N. Potapov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. N. Potapov.

Additional information

Original Russian Text © V.N. Potapov, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 1, pp. 54–63.

Supported in part by the Russian Foundation for Basic Research, project nos. 10-01-00616 and 11-01-00997, and Federal Target Program “Research and Educational Personnel of Innovation Russia” for 2009–2013, government contract no. 02.740.11.0362.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Potapov, V.N. Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes. Probl Inf Transm 48, 47–55 (2012). https://doi.org/10.1134/S003294601201005X

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S003294601201005X

Keywords

Search

Navigation