Let g$($x$)=$ x$^3+$ x $+1.$ Consider the information sequence $1101($i$.$e$.,$ i$($x$)=$x$^3+$ x$^2+1).$

$($i$)$ Find the $(7,4)$ codeword $($i$.$e$.,$ codeword is $7$ bits, where the message is $4$ bits$)$

corresponding to the preceding information sequence.

$($ii$)$ Suppose that the codeword is transmitted with no error. Demonstrate how the receiver extracts the message from codeword?

$($iii$)$ Prove that this generator enables to detect single bit errors.

$($iv$)$ Assuming that the system can detect $2-$bit errors, and let the receiver know that there is only a single bit error in the transmission. Can the receiver detect the error bit, and thus, obtain the correct message? Defend your answer.