Intersection patterns of linear subspaces with the hypercube

Following a combinatorial observation made by one of us recently in relation to a problem in quantum information [Nakata et al., Phys. Rev. X 7:021006 (2017)], we study what are the possible intersection cardinalities of a $k$-dimensional subspace with the hypercube in $n$-dimensional Euclidean space. We also propose two natural variants of the problem by restricting the type of subspace allowed. We find that whereas every natural number eventually occurs as the intersection cardinality for some $k$ and $n$, on the other hand for each fixed k, the possible intersections sizes are governed by severe restrictions. To wit, while the largest intersection size is evidently $2^k$, there is always a large gap to the second largest intersection size, which we find to be $\frac34 2^k$ for $k \geq 2$ (and $2^{k-1}$ in the restricted version). We also present several constructions, and propose a number of open questions and conjectures for future investigation.