Imagine the set of all binary vectors of length $2m$ where each of the vectors has $m$ ones and $m$ zeros.
I want to select some $n$ of these vectors such that the shortest distance among all pairs of vectors from this set of $n$ vectors is maximized. In other words I want a set of vectors which are maximally distant from each other.
I define distance by the number of positions at which the corresponding digits are different. For example distance between vectors $[1,0,0,1,1,0]$ and $[1,1,0,0,0,1]$ is $4$.
For example for $m=4$ and $n=2$ the solution can be $[1,1,1,1,0,0,0,0]$ and $[0,0,0,0,1,1,1,1]$.
How can I find a solution for arbitrary $m$ and $n$?