Maximum subset of words with Hamming distance ≥ D
In coding theory, the quantity you are looking for is called $A_q(n, d)$, where $n$ is the length of vectors, $d$ is the minimum distance between them, and $q$ is the alphabet size (omitted when $q=2$). Characterizing $A_q(n,d)$ is a challenging open problem (with many basic questions remaining unanswered) but various asymptotic and non-asymptotic upper and lower bounds are known. See Chapter 17 of the book "The Theory of Error-Correcting Codes" by MacWilliams and Sloane for a summary of the most important ones.