I want to send one bit $x$ over a noisy channel, specifically, a binary symmetric channel with error probability $p$, where $p=(1-\epsilon)/2$ and $\epsilon$ is small. In other words, the error probability is close to 1/2, so the capacity of the channel is small.
The obvious method is to use a repetition code: repeat $x$ many times. The receiver will then use majority vote to recover $x$.
Suppose I want an overall error probability of at most $\delta$ (i.e., the probability that the receiver guesses $x$ incorrectly is at most $\delta$). How many times do I need to repeat $x$?
If $\delta$ is fixed, I can see that the number of repetitions needs to be proportional to $1/(1-h_2(p))$, where $h_2$ is the binary entropy function. My question is: what is the dependence on $\delta$?
I'm guessing we should send $c_\delta / (1-h_2(p))$ copies of $x$, for constant $c_\delta$ that depends on $\delta$; how does $c_\delta$ depend on $\delta$, asymptotically, for small values of $\epsilon,\delta$? (For example, is it something like $c_\delta \sim 1 + \log 1/\delta$?)