Skip to main content
SpringerLink
Log in

The sextuply shortened binary Golay code is optimal

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

Abstract

The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions. Specifically, denoting the maximum size of an unrestricted binary code of length n and minimum distance d by A(n, d), it has been known that \(64 \le A(18,8) \le 68\) and \(128 \le A(19,8) \le 131\). In the current computer-aided study, it is shown that \(A(18,8)=64\) and \(A(19,8)=128\), so an optimal code is obtained even after shortening the extended binary Golay code six times.

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. Agrell E., Vardy A., Zeger K.: A table of upper bounds for binary codes. IEEE Trans. Inf. Theory 47, 3004–3006 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  2. Best M.R., Brouwer A.E., MacWilliams F.J., Odlyzko A.M., Sloane N.J.A.: Bounds for binary codes of length less than 25. IEEE Trans. Inf. Theory 24, 81–93 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  3. Brouwer A.E.: Disclosure. http://www.win.tue.nl/~aeb/codes/lpdetail.html.

  4. Delsarte P.: Bounds for unrestricted codes, by linear programming. Philips Res. Rep. 27, 272–289 (1972).

    MathSciNet  MATH  Google Scholar 

  5. Delsarte P., Goethals J.-M.: Unrestricted codes with the Golay parameters are unique. Discret. Math. 12, 211–224 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  6. Gijswijt D.C., Mittelmann H.D., Schrijver A.: Semidefinite code bounds based on quadruple distances. IEEE Trans. Inf. Theory 58, 2697–2705 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  7. Golay M.J.E.: Notes on digital coding. Proc. IRE 37, 657 (1949).

    Google Scholar 

  8. Hamming R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 29, 147–160 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  9. Johnson S.M.: On upper bounds for unrestricted binary error-correcting codes. IEEE Trans. Inf. Theory 17, 466–478 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  10. Kaski P., Östergård P.R.J.: Classification Algorithms for Codes and Designs. Springer, Berlin (2006).

    MATH  Google Scholar 

  11. Kim H.K., Toan P.T.: Improved semidefinite programming bound on sizes of codes. IEEE Trans. Inf. Theory 59, 7337–7345 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  12. Krotov D.S., Östergård P.R.J., Pottonen O.: On optimal binary one-error-correcting codes of lengths \(2^m-4\) and \(2^m-3\). IEEE Trans. Inf. Theory 57, 6771–6779 (2011).

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. McKay B.D.: Isomorph-free exhaustive generation. J. Algorithms 26, 306–324 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  15. McKay B.D., Piperno A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  16. Niskanen S., Östergård P.R.J.: Cliquer User’s Guide: Version 1.0, Technical report T48, Communications Laboratory, Helsinki University of Technology, Espoo (2003).

  17. Östergård P.R.J.: A fast algorithm for the maximum clique problem. Discret. Appl. Math. 120, 197–207 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  18. Östergård P.R.J.: On the size of optimal three-error-correcting binary codes of length 16. IEEE Trans. Inf. Theory 57, 6824–6826 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  19. Östergård P.R.J.: On optimal binary codes with unbalanced coordinates. Appl. Algebra Eng. Commun. Comput. 24, 197–200 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  20. Östergård P.R.J., Pottonen O.: The perfect binary one-error-correcting codes of length 15. I. Classification. IEEE Trans. Inf. Theory 55, 4657–4660 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  21. Plotkin M.: Binary Codes with Specified Minimum Distance, M.Sc. Thesis [cf. Refs. 25 & 26], Moore School of Electrical Engineering, University of Pennsylvania (1952).

  22. Plotkin M.: Binary codes with specified minimum distance. IRE Trans. Inf. Theory 6, 445–450 (1960).

    Article  MathSciNet  Google Scholar 

  23. Schrijver A.: New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Trans. Inf. Theory 51, 2859–2866 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  24. Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

  25. Snover S.L.: The Uniqueness of the Nordstrom–Robinson and the Golay Binary Codes, Ph.D. Thesis, Department of Mathematics, Michigan State University (1973).

  26. van Pul C.L.M.: On Bounds on Codes, M.Sc. Thesis, Department of Mathematics and Computer Science, Eindhoven University of Technology (1982).

Download references

Author information

Authors and Affiliations

  1. Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 15400, 00076, Aalto, Finland

    Patric R. J. Östergård

Authors
  1. Patric R. J. Östergård
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patric R. J. Östergård.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”

Supported in part by the Academy of Finland, Project #289002.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Östergård, P.R.J. The sextuply shortened binary Golay code is optimal. Des. Codes Cryptogr. 87, 341–347 (2019). https://doi.org/10.1007/s10623-018-0532-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-018-0532-z

Keywords

Mathematics Subject Classification

Search

Navigation