Let $A(n, d)$ denote the maximum size of a binary code of length $n$ with minimum distance $d$. For fixed $\delta$ with $0<\delta<1 / 2$, let $\alpha(\delta)=\limsup _{n} \frac{1}{n} \log _{2} A(n, n \delta)$. Show that

$$1-H(\delta) \leqslant \alpha(\delta) \leqslant 1-H(\delta / 2)$$

where $H(p)=-p \log _{2} p-(1-p) \log _{2}(1-p)$.

[You may assume the GSV and Hamming bounds and any form of Stirling's theorem provided you state them clearly.]