# Existence of good error correcting codes

I recently asked this question and got an answer from Yuval Filmus stating that we can build a solution using error-correcting codes.

More specifically, I'm looking for error correcting codes (for binary alphabet) with constant $$R>0$$ non-zero relative rate, and as high as possible relative distance $$\delta$$.

I know that using this theorem, we can achieve any $$\delta < \frac{1}{2}$$. As pointed out by Yuval Filmus's answer to my last question, this is the best $$\delta$$ we can hope for.

Where can I find a proof that states there is no binary error-correcting code with a relative distance bigger than $$\frac{1}{2}$$?