Advertisement

Journal of Applied and Industrial Mathematics

, Volume 6, Issue 4, pp 451–459 | Cite as

Affine nonsystematic codes

  • S. A. Malyugin
Article

Abstract

A perfect binary code C of length n = 2 k − 1 is called affine systematic if there exists a k-dimensional subspace of {0, 1} n such that the intersection of C and each coset with respect to this subspace is a singleton; otherwise C is called affine nonsystematic. In this article we construct affine nonsystematic codes.

Keywords

perfect code Hamming code nonsystematic code affine nonsystematic code component 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. V. Avgustinovich and F. I. Solov’eva, “On Nonsystematic Perfect Binary Codes,” Problemy Peredachi Inforatsii 32(3), 47–50 (1996) [Problems of Inform. Transm. 32 (3), 258–261 (1996)].MathSciNetGoogle Scholar
  2. 2.
    S. V. Avgustinovich and F. I. Solov’eva, “Construction of Perfect Binary Codes by Sequential Shifts of α-Components,” Problemy Peredachi Informatsii, 33(3), 15–21 (1997). [Problems of Inform. Transm. 33 (3), 202–207 (1997)].MathSciNetGoogle Scholar
  3. 3.
    S. A. Malyugin, “On a Criterion for the Perfect Binary Codes to be Nonsystematic,” Dokl. Akad. Nauk 375(1), 13–16 (2000).MathSciNetGoogle Scholar
  4. 4.
    S. A. Malyugin, “Nonsystematic Perfect Binary Codes,” Diskret. Anal. Issled. Oper. Ser. 1, 8(1), 55–76 (2001).MathSciNetMATHGoogle Scholar
  5. 5.
    S. A. Malyugin, “On Affine Nonsystematic Codes,” Proc. Intern. Conf. dedicated to A. A. Lyapunov 90 Anniversary (Novosibirsk, October 8–12 (2001). Novosibirsk: Institute of Mathematics, 2001. P. 393–394. (See http://www.sbras.nsc.ru/ws/Lyap2001/2288).Google Scholar
  6. 6.
    S. A. Malyugin, “On Enumeration of Nonequivalent Perfect Binary Codes of Length 15 and Rank 15,” Diskret. Anal. Issled. Oper. Ser. 1, 13(1), 77–98 (2006).MathSciNetMATHGoogle Scholar
  7. 7.
    A. M. Romanov, “On Nonsystematic Perfect Binary Codes of Length 15,” Diskret. Anal. Issled. Oper. Ser. 1, 4(4), 75–78 (1997) [Discrete Appl. Math. 135 (1), 255–258 (2004)].MathSciNetMATHGoogle Scholar
  8. 8.
    P. R. J. Östergard and O. Pottonen, “The Perfect Binary One-Error-Correcting Codes of Length 15. Part I: Classification,” IEEE Trans. Inform. Theory 55(10), 4657–4660 (2009).MathSciNetCrossRefGoogle Scholar
  9. 9.
    P. R. J. Östergard, O. Pottonen, and K. T. Phelps, “The Perfect Binary One-Error-Correcting Codes of Length 15. Part II: Properties,” IEEE Trans. Inform. Theory 56(6), 2571–2582 (2009).CrossRefGoogle Scholar
  10. 10.
    K. T. Phelps and M. J. LeVan, “Kernels of Nonlinear Hamming Codes,” Des. Codes Cryptography 6(3), 247–257 (1995).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    K. T. Phelps and M. J. LeVan, “Nonsystematic Perfect Codes,” SIAM J. Discrete Math. 12(1), 27–34 (1999).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    F. I. Solov’eva, “Switchings and Perfect Codes,” in Numbers, Information and Complexity, (Dordrecht, Kluwer Acad., 2000), pp. 311–324.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations