Representations of code loops by binary codes

Abstract

Code loops are Moufang loops constructed from doubly even binary codes. Then, given a code loop $L$, we ask which doubly even binary code $V$ produces $L$. In this sense, $V$ is called a representation of $L$. In this article we define and show how to determine all minimal and reduced representations of nonassociative code loops of rank $3$ and $4$.

30 pages, 19 figures. Revised and extended version of arXiv:1909.05143, following referee reports. Submitted to the International Journal of Algebra and Computation