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

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.