On the number of frequency hypercubes $\mathrm{F}^n(4;2,2)$

A frequency $n$-cube $\mathrm{F}^n(4;2,2)$ is an $n$-dimensional $4$-by-…-by-$4$ array filled by $0$s and $1$s such that each line contains exactly two $1$s. We classify the frequency $4$-cubes $\mathrm{F}^4(4;2,2)$, find a testing set of size $25$ for $\mathrm{F}^3(4;2,2)$, and derive an upper bound on the number of $\mathrm{F}^n(4;2,2)$. Additionally, for every $n$ greater than $2$, we construct an $\mathrm{F}^n(4;2,2)$ that cannot be refined to a Latin hypercube, while each of its sub-$\mathrm{F}^{n-1}(4;2,2)$ can.