Direct construction of code loops

Code loops were introduced by R.L. Griess Jr. [Code loops, J. Algebra 100 (1) (1986) 224–234] and T. Hsu [Explicit constructions of code loops as centrally twisted products, Math. Proc. Cambridge Philos. Soc. 128 (2) (2000) 223–232] gave methods to construct the corresponding code loop from any given doubly even binary code; both these methods used some kind of induction. In this paper, we present a global construction of the loop, where we apply the correspondence between the concepts of Moufang loops and groups with triality.