nLab code loop

Overview

A code loop is a certain sort of Moufang loop constructed as a central extension of certain vector spaces over the finite field $\mathbb{F}_2$. 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 1986). 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 $\mathbb{Z}/2$ (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 given by Hsu 2000.

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

• Robert Griess: Code loops, J. Algebra 100 1 (1986) 224-234 [doi:10.1016/0021-8693(86)90075-X]

• Tim Hsu: Explicit constructions of code loops as centrally twisted products, Math. Proc. Cambridge Philos. Soc. 128 2 (2000) 223-232 doi:10.1017/S030500419900403X]

• Tim Hsu: Moufang loops of class 2 and cubic forms, Math. Proc. Camb. Phil. Soc. 128 (2000) 197-222 [doi:10.1017/S0305004199003977, math.GR/9611214]

  (arXiv title: Class 2 Moufang loops, small Frattini Moufang loops, and code loops)

• Ben Nagy, David Roberts: (Re)constructing code loops, American Mathematical Monthly 128 2 (2021) 151–161 [doi:10.1080/00029890.2021.1852047, arXiv:1903.02748]