This is a coding theory question. You want to find a binary constant weight $m$ code with $k$ codewords, and of maximal possible distance. There was a lot of research done on this.

This is a coding theory question. You want to find a binary constant weight $m$ code with $k$ codewords, and of maximal possible distance. There was a lot of research done on this.

For the specific case of $k$ small compared to $n$, one can take sufficiently many copies of a nice constant weight code with $k$ words. E.g. for $k=14$ you can take the the indicator functions of the hyperplanes of the 3-dimensional affine space over $\mathbb{F}_2$ (which has 8 points). By taking this $n'$ times, this would give you for $n=8n'$ and $m=4n'$, 14 subsets of size $m$ that have either empty intersection, or intersection of size $2n'$.