Explicit constructions of code loops as centrally twisted products

Abstract

Code loops are certain Moufang loop extensions of doubly even binary codes which are useful in finite group theory (e.g. Conway's construction of the Monster). We give several methods for explicitly constructing code loops as centrally twisted products. More specifically, after establishing some preliminary examples, we show how to use decompositions of codes to build code loops out of more familiar pieces, such as abelian groups, extraspecial groups, or the octonion loop. In particular, we use Turyn's construction of the Golay code to give a simple explicit construction of the Parker loop, one which may have applications to the study of the Monster.