What is the channel matrix of a binary symmetric channel with error probability $p$ ?

State the maximum likelihood decoding rule and the minimum distance decoding rule. Prove that if $p<1 / 2$, then they agree.

Let $C$ be the repetition code $\{000,111\}$. Suppose a codeword from $C$ is sent through a binary symmetric channel with error probability $p$. Show that, if the minimum distance decoding rule is used, then the probability of error is $3 p^{2}-2 p^{3}$.