On some projective triply-even binary codes invariant under the Conway group {\rm Co}_۱

A binary triply-even [۹۸۲۸۰, ۲۵, ۴۷۱۰۴]_۲ code invariant under the sporadic simple group {\rm Co}_۱ is constructed by adjoining the all-ones vector to the faithful and absolutely irreducible ۲۴-dimensional code of length ۹۸۲۸۰. Using the action of {\rm Co}_۱ on the code we give a description of the nature of the codewords of any non-zero weight relating these to vectors of types ۲, ۳ and ۴, 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}_۱. Moreover, we give a partial description of the nature of the codewords of minimum weight of the dual code.

Keywords:

automorphism group , modular representation , Conway group