# (Re)constructing Code Loops

## TL;DR

This paper explores an experimental method for reconstructing the multiplication structure of code loops, particularly Parker's loop, using partial cocycle data and algebraic identities, revealing new insights into their structure.

## Contribution

It introduces a novel approach to reconstruct code loops from limited cocycle information, enhancing understanding of their algebraic structure.

## Key findings

- Reconstructed Parker's loop multiplication from partial cocycle data
- Identified large subspaces where Parker's loop splits as a direct product
- Demonstrated the effectiveness of algebraic identities in computing code loops

## 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.