This question is motivated by section 15.1 (Codes) of Alon and Spencer's The probabilistic method.
Fix $\alpha<\frac{1}{2}$ and for each $n\in\mathbb{N}$ let $\{0,1\}^n$ be the length $n$ binary strings with the Hamming distance.
Let $N_n$ be the maximal cardinality of a $\alpha n$-separated set (that is, the points of the set are pairwise at distance $\alpha n$) in $\{0,1\}^n$. What do we know about the growth rate of $N_n$?
It has to grow slower than $\frac{2^n}{\#B(0,\frac{n\alpha}{2})}$. So, as $\#B(0,\frac{n\alpha}{2})=2^{n(H(\frac{\alpha}{2})+o(1))}$ (where $H(x)=-x\log_2(x)-(1-x)\log_2(1-x)$), as proved in the section mentioned above of Alon and Spencer's book, this gives an upper bound of $2^{n(1-H(\frac{\alpha}{2})+o(1))}$. I don't think I have any decent lower bounds.
Best Answer
The best upper bound will depend on $\alpha.$ and also whether you are interested in asymptotics.
There is a nice short chapter describing some of what is known https://algo.epfl.ch/_media/en/courses/2008-2009/mct_l03.pdf. The quantity $\alpha_2(x)$ is what you are interested in. In general unless $\alpha$ is very small the best upper bound is the so-called MRRW (4 Americans) bound of McEliece-Rudemich-Rumsey-Welch obtained by linear programming.
The lower bound given is Gilbert-Varshamov, and all the others are upper bounds. There have been some limited improvements on it, see Jiang and Vardy which I believe has since appeared in the IEEE Transactions on Information Theory after Alex Vardy unexpectedly passed away.