  We consider a family, $\mathcal{F}$, of subsets of an $n$-set such that the
cardinality of the symmetric difference of any two elements
$F,F'\in\mathcal{F}$ is not a multiple of 4. We prove that the maximal size of
$\mathcal{F}$ is bounded by $2n$, unless $n\equiv{}3\mod4$ when it is bounded
by $2n+2$. Our method uses cubic forms on $\mathbb{F}_2^n$ and the
Hurwitz-Radon theory of square identities. We also apply this theory to obtain
some information about boolean cubic forms and so-called additive quadruples.
