(Re)constructing Code Loops. Issue 2 (27th January 2021)

• Record Type: Journal Article
• Authors: Nagy, Ben; Roberts, David Michael
• Is Part Of: American Mathematical Monthly. Volume 128:Issue 2(2021)
• Page Start: 151
• Page End: 161
• Publication Date: 2021-01-27
• Subjects: MSC: Primary 20N05 -- Secondary 20B40 -- 20G10; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2021.1852047
• Languages: English
• ISSNs: 0002-9890

Abstract

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.