Let $d \geqslant 2$. Define the Hamming code $C$ of length $2^{d}-1$. Explain what it means to be a perfect code and show that $C$ is a perfect code.

Suppose you are using the Hamming code of length $2^{d}-1$ and you receive the message $111 \ldots 10$ of length $2^{d}-1$. How would you decode this message using minimum distance decoding? Explain why this leads to correct decoding if at most one channel error has occurred.