Representations of code loops by binary codes

TL;DR

This paper investigates how nonassociative code loops of rank 3 and 4 can be represented by minimal and reduced doubly even binary codes, clarifying their structural relationships.

Contribution

It defines and classifies all minimal and reduced representations of nonassociative code loops of ranks 3 and 4.

Findings

All minimal representations of rank 3 and 4 code loops are characterized.

All reduced representations of these code loops are determined.

The relationship between code loops and their binary code representations is clarified.

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 determine all minimal and reduced representations of nonassociative code loops of rank 3 and 4.