You and a friend are using the C$(7,4)$ Hamming code to send some $4-$bit messages to

each other.

$($a$)$ You encode the message $0001$ and send the encoded $7-$bit sequence to your friend, who receives

$0011101.$ How many errors were introduced during transmission?

$($b$)$ You subsequently receive the encoded sequence $1101001$ from your friend. Assuming at most

one error, what is the $4-$bit message that your friend sent?