Minimally intersecting subsets of fixed size The question I have, is how to generate the following collection of subsets:
Given a set $S$ of size $n$. I want to find a sequence of $k$ subsets of fixed size $m$, $0<m<n$, such that at each stage $i$ in the sequence ($1 < i \leq k$), the largest intersection between any such pair of subsets, is as small as possible.
Any guidance is appreciated.
Edit:
Given the formulation by Dima Pasechnik below, I would like to add an example of such a sequence of subsets, or codewords:
1 1 1 1 0 0 0 0
0 0 0 0 1 1 1 1
0 0 1 1 1 1 0 0
1 1 0 0 0 0 1 1
1 0 1 0 1 0 1 0
0 1 0 1 0 1 0 1
In this example the number of codewords $k=6$, their weight $m=4$, and $n=8$. As can be seen, the largest pairwise Hamming distance decreases as we continue the sequence (which is fine for me). 
My interest lies in finding an algorithm that constructs such a sequence in an efficient way for any non-degenerate choice of $n,k,m$.
(I am aware that minimal intersecting subsets  is a related question, but not the same.)
 A: 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 $2n'$. 
A: maybe you would be interested in the "Minimal Overlapping Algorithm" presented in "Two New Combinatorial Problems involving
Dominating Sets for Lottery Schemes - Werner R Grundlingh" (at page 155):
The Algorithm has been transformed in a vba code and is downloadable from the forum "http://wheels.forumcommunity.net/" using this link:
http://wheels.forumcommunity.net/?act=Attach&type=post&id=401682134
