$4.(25$ points$)$ Assume the codeword transmitted is received as: T$0($X$)=$ X$20+$ X$13+$ X$12+$ X$11+$ X$8+$ X$7+$ X$2+$ X$,$ the error polynomial is: E$($X$)=$ X$19+$ X$13+$ X$7+$ X$3,$ and the generator polynomial is: G$($X$)=$ X$4+$ X$3+$ X $+1.$

a$.$ What was the original codeword transmitted? Show your answer in polynomial form.

b$.$ How many bits of errors have occurred?

c$.$ What is the size of the burst error?

d$.$ Is the receiver able to detect every burst error of this size $($obtained in part c$)?$ Explain clearly.