Representations of code loops by binary codes

TL;DR

This paper investigates how to identify minimal and reduced binary code representations of nonassociative code loops of rank 3 and 4, enhancing understanding of their structure.

Contribution

It introduces methods to determine all minimal and reduced representations of nonassociative code loops of specific ranks.

Findings

Complete characterization of minimal representations for rank 3 and 4 loops

Algorithms for constructing reduced representations

Insights into the structure of nonassociative code loops

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 $4$.