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

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## 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## 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.

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