Consider the binary code

$C$:$=\{ubrace(000000$ubrace${)}_{?}=c_1,$ubrace$(011110$ubrace${)}_{?}=c_2,$ubrace$(101101$ubrace${)}_{?}=c_3,$ubrace$(110011$ubrace${)}_{?}=c_4,$ubrace$(111000$ubrace${)}_{?}=c_5\}.$

$($I$)$ What is the minimum distance of $C?$ Verify your answer.

$($II$)$ What is the biggest number of errors in a codeword that can always be detected? Explain why.

$($III$)$ What is the biggest number of errors in a codeword that can always be corrected? Explain why.

$($IV$)$ If it is known that the word $101001$ contains at most one error, what was the original codeword?

$($V$)$ Why can a binary code with $5$ codewords $($such as $C)$ never be a linear code?