  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
$512$.
