Advertisement

Graphs and Combinatorics

, Volume 34, Issue 3, pp 509–521 | Cite as

A Projective Two-Weight Code Related to the Simple Group \(\mathrm{Co}_1\) of Conway

  • B. G. Rodrigues
Original Paper
First Online:
Received:
Revised:
  • 45 Downloads

Abstract

A binary \([98280, 24, 47104]_2\) projective two-weight code related to the sporadic simple group \(\mathrm{Co}_1\) of Conway is constructed as a faithful and absolutely irreducible submodule of the permutation module induced by the primitive action of \(\mathrm{Co}_1\) on the cosets of \(\mathrm{Co}_2\). The dual code of this code is a uniformly packed \([98280, 98256,3]_2\) code. The geometric significance of the codewords of the code can be traced to the vectors in the Leech lattice, thus revealing that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of \(\mathrm{Co}_1\). Similarly, the stabilizer of the codewords of minimum weight in the dual code is a maximal subgroup of \(\mathrm{Co}_1\). As by-product, a new strongly regular graph on 16777216 vertices and valency 98280 is constructed using the codewords of the code.

Keywords

Strongly regular graph Symmetric design Automorphism group Modular representation Conway group 

Mathematics Subject Classification

Primary 05B05 20D08 94B05 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

Part of this work was done while visiting the School of Mathematics and Statistics at the University of Sydney. The author wishes to express his sincere thanks to the Magma group, in particular John Cannon for splendid hospitality and financial support. The author also extends his gratitude to Bill Unger whose invaluable help with computations made this article possible and to Markus Grassl for computing the full weight distribution of the dual code and for suggesting the alternative proof given in Remark 4.4. The author is grateful to the anonymous referees for their valuable comments and suggestions which improved significantly the quality of the paper.

References

  1. 1.
    Atlas of finite group representations. http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/
  2. 2.
    Assmus, E.F. Jr., Key, J.D.: Designs and their Codes. Cambridge Tracts in Mathematics, vol. 103 (Second printing with corrections, 1993). Cambridge University Press, Cambridge (1992)Google Scholar
  3. 3.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brouwer, A.E., van Eijl, C.J.: On the \(p\)-rank of the adjacency matrices of strongly regular graphs. J. Algebraic Comb. 1, 329–346 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Calderbank, R., Kantor, W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Conway, J.H.: A group of order 8,315,553,613,086,720,000. Bull. Lond. Math. Soc. 1, 79–88 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: An Atlas of Finite Groups. Oxford University Press, Oxford (1985)zbMATHGoogle Scholar
  8. 8.
    Curtis, R.T.: On subgroups of O. I. lattice stabilizers. J. Algebra. 4, 549–573 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ganief, M.S.: 2-Generations of the Sporadic Simple Groups. PhD Thesis, University of Natal (1997)Google Scholar
  10. 10.
    Jansen, C.: The minimal degrees of faithful representations of the sporadic simple groups and their covering groups. LMS J. Comput. Math. 8, 122–144 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Key, J.D., Moori, J.: Designs, codes and graphs from the Janko groups \({J}_1\) and \({J}_2\). J. Comb. Math. Comb. Comput. 40, 143–159 (2002)zbMATHGoogle Scholar
  12. 12.
    Key, J.D., Moori, J., Rodrigues, B.G.: On some designs and codes from primitive representations of some finite simple groups. J. Comb. Math. Comb. Comput. 45, 3–19 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Key, J.D., Moori, J.: Correction to: Codes, designs and graphs from the Janko groups \(J_1\) and \(J_2\). J. Comb. Math. Comb. Comput. 40, 143–159 (2002)zbMATHGoogle Scholar
  14. 14.
    Key, J.D., Moori, J.: Correction to: Codes, designs and graphs from the Janko groups \(J_1\) and \(J_2\). J. Comb. Math. Comb. Comput. 64, 153 (2008)zbMATHGoogle Scholar
  15. 15.
    Knapp, W., Schmid, P.: Codes with prescribed permutation group. J. Algebra 67, 415–435 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Knapp, W., Schaeffer, Hans-J.: On the codes related to the Higman-Sims graph. Electron. J. Combin. 22(1), 58 (2015)Google Scholar
  17. 17.
    Liebeck, M.W., Praeger, C.E., Saxl, J.: The maximal factorizations of the finite simple groups and their automorphism groups. Mem. Am. Math. Soc. 86(432), 1–151 (1990)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Moori, J., Rodrigues, B.G.: A self-orthogonal doubly even code invariant under \({{\rm M}}^{\rm c}{\rm L}:2\). J. Comb. Theory Ser. A 110(1), 53–69 (2005)CrossRefzbMATHGoogle Scholar
  19. 19.
    Moori, J., Rodrigues, B.G.: Some designs and codes invariant under the simple group \({{\rm Co}}_2.\). J. Algebra 316, 649–661 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Moori, J., Rodrigues, B.G.: Some designs and binary codes preserved by the simple group \({{\rm Ru}}\) of Rudvalis. J. Algebra 372, 702–710 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Praeger, C.E., Soicher, L.H.: Low rank representations and graphs for sporadic groups. Australian Mathematical Society Lecture Series, vol. 8. Cambridge University Press, Cambridge (1997)Google Scholar
  22. 22.
    van Lint, J.: Introduction to Coding Theory. Graduate Texts in Mathematics, vol. 86, 3rd edn. Springer-Verlag London Ltd., London (1999)Google Scholar
  23. 23.
    Wilson, R.A.: The maximal subgroups of Conway’s group \({\rm Co}_1\). J. Algebra 85, 144–165 (1983)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wilson, R.A.: Vector stabilizers and subgroups of Leech lattice groups. J. Algebra 127, 387–408 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wilson, R.A.: The finite simple groups. Graduate Texts in Mathematics, vol. 251. Springer-Verlag London Ltd., London (2009)Google Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer ScienceUniversity of KwaZulu-NatalDurbanSouth Africa

Personalised recommendations

Cite article

Buy options