4 - The Binary Golay Code

Summary

The binary Golay code C is defined as the 12-dimensional vector space over Z2 spanned by the 759 octads interpreted as vectors with eight 1s and 16 0s. The MOG is constructed by considering two 3-dimensional spaces over Z2, the Point space and the Line space, whose codewords are of length 8, and gluing three copies together in such a way as to obtain a 12-dimensional subspace of the 24-dimensional space P(Ω), consisting of all subsets of Ω. The minimal weight codewords in this 24-dimensional space are shown to have weight 8 and to total 759. The construction thus proves that a Steiner system S(5, 8, 24) exists, and provides a unique label for each codeword in the binary Golay code. We exhibit a natural isomorphism between the 24-dimensional space P(Ω) factored by C and the dual space C⋆, and identify its elements as 24 monads, 276 duads, 2024 triads and (244)/6=1771 sextets; this last division by 6 occurs because two tetrads 4 whose union is an octad are congruent modulo C.