On some projective triply-even binary codes invariant under the Conway group ${\rm Co}_1$

Document Type : Research Paper

Author

Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa

Abstract

A binary triply-even $[98280, 25, 47104]_2$ code invariant under the sporadic simple group ${\rm Co}_1$ is constructed by adjoining the all-ones vector to the faithful and absolutely irreducible 24-dimensional code of length 98280. Using the action of ${\rm Co}_1$ on the code we give a description of the nature of the codewords of any non-zero weight relating these to vectors of types 2, 3 and 4, respectively of the Leech lattice. We show that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of ${\rm Co}_1$. Moreover, we give a partial description of the nature of the codewords of minimum weight of the dual code.

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References

[1] M. Aschbacher, Sporadic groups, Cambridge Tracts in Mathematics, 104, Cambridge University Press, Cambridge,
1994.
[2] E. F. Assmus, Jr and J. D. Key, Designs and their codes. Cambridge Tracts in Mathematics, 103, Cambridge University
Press, Cambridge, 1992.
[3] K. Betsumiya and A. Munemasa, On triply even binary codes, J. Lond. Math. Soc., 86 (2012) 1–16.
[4] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24
(1997) 235–265.
[5] J. H. Conway, A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc., 1 (1969) 79–88.
[6] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of finite groups, Maximal subgroups
and ordinary characters for simple groups, With computational assistance from J. G. Thackray. Oxford University
Press, Eynsham, 1985.
[7] D. Heinlein, T. Honold, M. Kermaier, S. Kurz and A. Wassermann, Projective divisible binary codes, In: The Tenth
International Workshop on Coding and Cryptography 2017 : WCC Proceedings. Saint-Petersburg, 2017.
[8] T. Honold, M. Kermaier, S. Kurz and A. Wassermann, The lengths of projective triply-even binary codes, IEEE Trans.
Inform. Theory, 66 (2020) 2713–2716.
[9] W. Knapp, Private communication.
[10] B. G. Rodrigues, A projective two-weight code related to the simple group Co1 of Conway, Graphs and Combin., 34
(2018) 509 – 521.
[11] R. A. Wilson, The maximal subgroups of Conway’s group Co1 , J. Algebra, 85 (1983) 144–165.
[12] R. A. Wilson, Vector Stabilizers and Subgroups of Leech Lattice Groups, J. Algebra, 127 (1989) 387–408.
[13] R. A. Wilson, The finite simple groups, Graduate Texts in Mathematics, 251, Springer-Verlag London, Ltd., London,
2009.

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