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 $\mathbb{Z}/2$ (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).

Tim Hsu, Explicit constructions of code loops as centrally twisted products, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 2, 223–232 (journal);

Tim Hsu, Moufang loops of class 2 and cubic forms, Math. Proc. Camb. Phil. Soc. 128 (2000), 197-222 (on the arXiv as math.GR/9611214 under the title Class 2 Moufang loops, small Frattini Moufang loops, and code loops).

Last revised on November 21, 2018 at 04:53:49.
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