Let $\Omega = \left\lbrace x : |x| = r \, \text{for} \, x \in \mathbb{Z}\_2^n \right\rbrace$ for $r \in \mathbb{N}$. We want to find the the biggest subset of $\Omega$, $\Gamma = \left\lbrace x \in \Omega : |x - y| \geq m ,\, \forall x, y \in \Gamma \right\rbrace$ for a given value of $m \in \mathbb{N}$.



Find the largest subset of all binary arrays of length $n$ with $r$ ones which have pairwise distance greater than $m$
What you are looking for is precisely the optimal (largest cardinality) constant weight (this weight is $r$ in your case) binary code with length $n$ and distance $m$, which is a well-researched and very difficult problem in general.


Let this quantity be denoted $A(n,m,r)$ in your terminology. In fact normally, this is denoted $A(n,d,w)$ with $d$ the minimum distance and $w$ the constant weight. There are tables of upper (see [here](https://www.win.tue.nl/~aeb/codes/Andw.html)) and lower (see [here](http://codes.se/bounds/)) bounds for small values of the parameters. 


Clearly, a constant weight code in general will have fewer codewords than an unrestricted code. So, general upper bounds on code cardinality will also upper bound a constant weight code.


Let $A(n,d)$ be the largest possible number of codewords in such an unrestricted binary code with length $n$ and minimum distance $d.$ Then by the fact that $r-$spheres around codewords must be disjoint where $r=\lfloor (d-1)/2\rfloor,$
such a code $\Omega$ must obey
$$
\#\Omega\leq \frac{2^n}{\sum\_{k=0}^r \binom{n}{k}}
$$
where the denominator is the volume of the Hamming sphere of radius $r.$


