Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 341–347 | Cite as

The sextuply shortened binary Golay code is optimal

  • Patric R. J. ÖstergårdEmail author
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


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(nd), 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.


Classification Clique Double counting Error-correcting code Golay code 

Mathematics Subject Classification

94B25 94B65 90C27 



  1. 1.
    Agrell E., Vardy A., Zeger K.: A table of upper bounds for binary codes. IEEE Trans. Inf. Theory 47, 3004–3006 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
  4. 4.
    Delsarte P.: Bounds for unrestricted codes, by linear programming. Philips Res. Rep. 27, 272–289 (1972).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Delsarte P., Goethals J.-M.: Unrestricted codes with the Golay parameters are unique. Discret. Math. 12, 211–224 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gijswijt D.C., Mittelmann H.D., Schrijver A.: Semidefinite code bounds based on quadruple distances. IEEE Trans. Inf. Theory 58, 2697–2705 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Golay M.J.E.: Notes on digital coding. Proc. IRE 37, 657 (1949).Google Scholar
  8. 8.
    Hamming R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 29, 147–160 (1950).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Johnson S.M.: On upper bounds for unrestricted binary error-correcting codes. IEEE Trans. Inf. Theory 17, 466–478 (1971).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kaski P., Östergård P.R.J.: Classification Algorithms for Codes and Designs. Springer, Berlin (2006).zbMATHGoogle Scholar
  11. 11.
    Kim H.K., Toan P.T.: Improved semidefinite programming bound on sizes of codes. IEEE Trans. Inf. Theory 59, 7337–7345 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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).CrossRefzbMATHGoogle Scholar
  13. 13.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).zbMATHGoogle Scholar
  14. 14.
    McKay B.D.: Isomorph-free exhaustive generation. J. Algorithms 26, 306–324 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    McKay B.D., Piperno A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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).Google Scholar
  17. 17.
    Östergård P.R.J.: A fast algorithm for the maximum clique problem. Discret. Appl. Math. 120, 197–207 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Östergård P.R.J.: On optimal binary codes with unbalanced coordinates. Appl. Algebra Eng. Commun. Comput. 24, 197–200 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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).Google Scholar
  22. 22.
    Plotkin M.: Binary codes with specified minimum distance. IRE Trans. Inf. Theory 6, 445–450 (1960).MathSciNetCrossRefGoogle Scholar
  23. 23.
    Schrijver A.: New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Trans. Inf. Theory 51, 2859–2866 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).Google Scholar
  25. 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).Google Scholar
  26. 26.
    van Pul C.L.M.: On Bounds on Codes, M.Sc. Thesis, Department of Mathematics and Computer Science, Eindhoven University of Technology (1982).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Communications and NetworkingAalto University School of Electrical EngineeringAaltoFinland

Personalised recommendations