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][1]. 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][2] which I believe has since appeared in the IEEE Transactions on Information Theory after Alex Vardy unexpectedly passed away.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/ffd8x.jpg
  [2]: https://arxiv.org/pdf/math/0404325.pdf