Code loops in dimension at most 8

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 $512$.