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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