Code loops in dimension at most 8

TL;DR

This paper classifies and enumerates code loops of dimension up to 8, revealing the vast number of such loops and their structure, which are important in the context of sporadic groups and algebraic combinatorics.

Contribution

It provides a complete enumeration of code loops of dimension at most 8 using classifications of trilinear alternating forms over GF(2).

Findings

767 code loops of order 128

80826 code loops of order 256

937,791,557 code loops of order 512

Abstract

Code loops are certain Moufang $2$-loops constructed from doubly even binary codes that play an important role in the construction of local subgroups of sporadic groups. More precisely, code loops are central extensions of the group of order $2$ by an elementary abelian $2$-group $V$ in the variety of loops such that their squaring map, commutator map and associator map are related by combinatorial polarization and the associator map is a trilinear alternating form. Using existing classifications of trilinear alternating forms over the field of $2$ elements, we enumerate code loops of dimension $d=\mathrm{dim}(V)\le 8$ (equivalently, of order $2^{d+1}\le 512$) up to isomorphism. There are $767$ code loops of order $128$, and $80826$ of order $256$, and $937791557$ of order $512$.