Using the previous problem, prove that, for any 1 \leq d \leq k, any binary encoding map with message length k and minimum distance d has block length n \geq k+\frac{d-2}{2} \log _{2}\left(\frac{k}{d}\right). You may use the fact that \binom{n}{t} \geq\left(\frac{n}{t}\right)^{t} for any positive integers n and t with n \geq t.