  We compute a complete set of isomorphism classes of cubic fourfolds over
$\mathbb{F}_2$. Using this, we are able to compile statistics about various
invariants of cubic fourfolds, including their counts of points, lines, and
planes; all zeta functions of the smooth cubic fourfolds over $\mathbb{F}_2$;
and their Newton polygons. One particular outcome is the number of smooth cubic
fourfolds over $\mathbb{F}_2$, which we fit into the asymptotic framework of
discriminant complements. Another motivation is the realization problem for
zeta functions of $K3$ surfaces. We present a refinement to the standard method
of orbit enumeration that leverages filtrations and gives a significant
speedup. In the case of cubic fourfolds, the relevant filtration is determined
by Waring representation and the method brings the problem into the
computationally tractable range.
