We are considering length n strings over the alphabet S.
The goal is to compute a set of string, M, such that for any string, x, in S^n there exists a string, y, in M such that the hamming distance between x and y is at least r.
A way of thinking of this is that we want to guarantee that for any string, we have a string in our solution M which is far away from the given string.
For a given S (with size at least 3), n and r (which may be a funciton of n, for example n/2) for many strings a needed to include in M to guarantee that for any string in S^n, we have a string in M which has a hamming distance to it of at least r.
I have a feeling this is well studied and might be related to covering designs. Can someone give me a good reference (or insight)? I would be very grateful.