$$1-H(\delta) \le R(\delta) \le H(\frac{1}{2} -\sqrt {\delta (1-\delta)}).$$

This formula describes the state-of-the-art lower and upper bound for the rate of binary codes of length $n$, as $n$ tends to infinity, and minimal distance $\delta n$, $0 < \delta < 1/2$. The lower bound is due to Gilbert (1952). Now better lower bound is known today. The upper bound is by McEliece, Rudemich, Rumsey, and Welsh (MRRW) (1977), who described also an improved upper bound for $\delta \ge 0.273$. No better upper bounds than those discovered by MRRW are known today.

Van Lint's book on the theory of error-correcting codes is a good source. (*)

$$1-H(\delta) \le R(\delta) \le H\left(\frac{1}{2} -\sqrt {\delta (1-\delta)}\right).$$

This formula describes the state-of-the-art lower and upper bound for the rate of binary codes of length $n$, as $n$ tends to infinity, and minimal distance $\delta n$, $0 < \delta < 1/2$. The lower bound is due to Gilbert (1952). No better lower bound is known today. The upper bound is by McEliece, Rudemich, Rumsey, and Welsh (MRRW) (1977), who described also an improved upper bound for $\delta \ge 0.273$. No better upper bounds than those discovered by MRRW are known today.

Van Lint's book on the theory of error-correcting codes is a good source. (*)