Intersection sizes of linear subspaces with the hypercube

We continue the study by Melo and Winter [arXiv:1712.01763, 2017] on the possible intersection sizes of a $k$-dimensional subspace with the vertices of the $n$-dimensional hypercube in Euclidean space. Melo and Winter conjectured that all intersection sizes larger than $2^{k-1}$ (the "large" sizes) are of the form $2^{k-1}+2^i$. We show that this is almost true: the large intersection sizes are either of this form or of the form $35\cdot 2^{k-6}$. We also disprove a second conjecture of Melo and Winter by proving that a positive fraction of the "small" values is missing.