Let $A=\{0,1\}^n$ with distance $d(x,y)=\sum_i |x_i-y_i|$ (i.e. for example with $n=2$ we have $d([0,1],[1,0])=2$).
Given $m<n$ how well can we "approximate $A$" with a subset of cardinality $m$?
Formally, I want to know
$$g(m):=\inf_{S\subset A,\ \#S=m} \sup_{x \in A} d(x,S)$$
I tried to do it by myself, founding some trivial cases $$g(1)=n$$ $$g(2)=\frac{n}{2}$$
but then I can't proceed. Also I think that such a simple problem must have been treated by someone before me