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We consider a family, $\mathcal{F}$, of subsets of an $n$-set such that thecardinality 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 boundedby $2n+2$. Our method uses cubic forms on $\mathbb{F}_2^n$ and theHurwitz-Radon theory of square identities. We also apply this theory to obtainsome information about boolean cubic forms and so-called additive quadruples.

Extremal set theory, cubic forms on $\mathbb{F}_2^n$ and Hurwitz square identities