Q$4)$ A linear cyclic code $(n,k)$ has code length $n=7$ and generator polynomial

$g(x)=1+x^2+x^3+x^4$

$($a$)$ Find the code rate, the generator and parity check matrices of the code in systematic form, and its Hamming distance.

$($b$)$ If all the information symbols are $\text{'}1\text{'}$s$,$ what is the corresponding code vector?

$($c$)$ Find the syndrome corresponding to an error in the first information symbol, and show that the code is capable of correcting this error.