(Re)constructing Code Loops
arXiv:1903.02748 · doi:10.1080/00029890.2021.1852047
Abstract
The Moufang loop named for Richard Parker is a central extension of the extended binary Golay code. It the prototypical example of a general class of nonassociative structures known today as code loops, which have been studied from a number of different algebraic and combinatorial perspectives. This expository article aims to highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops' twisted cocycles satisfy. As a byproduct we demonstrate that one can reconstruct the multiplication in Parker's loop from a mere fragment of its twisted cocycle. We also give relatively large subspaces of the Golay code over which Parker's loop splits as a direct product.
arXiv comment10 pages, three figures, one ancillary text file containing enough data to reconstruct the multiplication in the Parker loop. v2 Updated after referee feedback