nLab
code loop

Code loops

Overview

A code loop is a certain sort of Moufang loop constructed as a central extension of certain vector spaces over the finite field 𝔽 2\mathbb{F}_2. In particular, such vector spaces are taken to be doubly even binary linear codes.

A general result about the existence of code loops is as follows.

Theorem

(Griess). Every doubly even binary code has a unique extension by a non-associative Moufang loop.

This can be seen as a result about the cohomology used to define code loops, namely that it is isomorphic to ℤ/2\mathbb{Z}/2 (one always has the trivial extension, which is just a group).

Code loops were originally defined independently by Robert Griess and Richard Parker. A modern description is in (Hsu).

Parker’s code loop was used by John Conway to construct the Monster group, and arises from taking the binary code to be the binary Golay code.

References