Keep in mind I know virtually no coding theory, I simply recognized this as an equivalent formulation to another question I was considering.
Let $F(r,n)$ denote the set of all words of length $r$ on $n$ letters. and let $H(f,g)$ be the Hamming distance between $f$ and $g$.
What I wish to know is the following:
For fixed integers $r,n,k$ what is the maximum possible cardinality of $S \subset F(r,n)$ with the property that $ \forall f,g \in S$, $H(f,g) \ge k$.
Any information you can give me about this would be greatly appreciated, thanks!
In general your question has no complete answer. Even if you are in the case that $n=q$ is a prime power the answer is very hard in general.
However, when the size of the alphabet is a prime power you could have a look at some bounds on the maximum possible cardinality of $S$ here
Some upper bounds
https://en.wikipedia.org/wiki/Hamming_bound
https://en.wikipedia.org/wiki/Griesmer_bound
https://en.wikipedia.org/wiki/Singleton_bound
https://en.wikipedia.org/wiki/Johnson_bound
A lower bound
https://en.wikipedia.org/wiki/Gilbert%E2%80%93Varshamov_bound