What is the size of the largest subset with a pairwise hamming distance of 3

Question

Consider all binary strings of length $n$. Is there any known bounds on the size of the maximum subset such that the pairwise hamming distance between any two elements is at least 3.

Answer 1

Edit: I found a much better solution, so I have overhauled this answer.

You already have the upper bound $$A_2(n,3)\le \frac{2^{n}}{n+1}.$$ which is just the sphere bound. For a lower bound, you can use the Hamming code of length $n$, which proves $$ A_2(n,3)\ge 2^{n-\lfloor \log_2(n)\rfloor-1}\ge \frac{2^{n-2}}{n} $$ Therefore, this is within a factor of $4$ of the optimal code.

In general, finding the optimal code is an open problem. There is a table of best known bounds for binary codes maintained here.