In the book "Essential Coding Theory" by Venkatesan Guruswami, Atri Rudra, and Madhu Sudan, in the Zyablov bound section, they claim that when the relative distance $\delta = \frac{1}{2} - \epsilon$, the Zyablov bound states that there is a code with rate $\geq \Omega(\epsilon^3)$. How can we show this?
Some facts that I have tried to use are that $H(1/2 - \epsilon) \leq 1 - c_q\epsilon^2$ (which was useful to prove a similar statement for the GV bound), and $H^{-1}\left(y - \frac{\epsilon^2}{c_q'}\right) \geq H^{-1}(y) - \epsilon$, which is lemma 3.3.7 from the book. But I haven't been able to figure the optimization out.
Any help or pointers would be greatly appreciated!