Define the Hamming code $h: \mathbb{F}_{2}^{4} \rightarrow \mathbb{F}_{2}^{7}$ and prove that the minimum distance between two distinct code words is 3. Explain how the Hamming code allows one error to be corrected.

A new code $c: \mathbb{F}_{2}^{4} \rightarrow \mathbb{F}_{2}^{8}$ is obtained by using the Hamming code for the first 7 bits and taking the last bit as a check digit on the previous 7 . Find the minimum distance between two distinct code words for this code. How many errors can this code detect? How many errors can it correct?