The polynomial $x^4-1$ factorises to $(x-1)(x+1)(x^2+1).$ Consider a cyclic code built using $g(x)=x+1.$

$($a$)$ Generate a table of the messages and corresponding $($non$-$systematic$)$ codewords from this code.

$(10$ marks$)$

$($b$)$ What is the Hamming distance of this code and hence, how many errors can this code correct?

$(2$ marks$)$

$($c$)$ Using your answer in $($b$),$ determine if this code is useful. Explain your answer.

$(3$ marks$)$

$($d$)$ Now consider a cyclic code with $g(x)=x^2+1.$ Encode the message $011$ in systematic form, giving your answer in binary format.