I'm working on an application for which I want to take the set $C$ of all the possible $k$-combinations of elements in $M$ (with $||M|| = m$), and cover $C$ with the sets of $k$-combinations of subsets $N\_i$ of $M$, with $||N\_i|| = n < m$ $\forall$ $N\_i$ 


So there are $m \choose k$ combinations to cover, and each set $Q\_i$ of $n$ elements will contain $n \choose k$ combinations. 


What I'd like is an algorithm that constructs the sets $Q\_i$ such that $q$ is minimized (i.e., as close to $m \choose k$ $/$ $n \choose k$ as possible)


So, for example, if $m = 100$, $k = 3$, and $n = 10$, I would want the smallest set of sets of $10$ elements such that their respective sets of $3$-combinations covered the set of $100 \choose 3$ $3$-combinations of M. 



how to cover set of k-combinations of M with subsets of M
The magic words are "design" (if the minimum can be realized) and "covering design" otherwise. There is a vast literature and no algorithm.


