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 here. 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-Rumsey-Rudemich-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.

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.