$($a$)$ Explain the working of parity, checksum and CRC error chocking schemes.

$($b$)$ Why are CRC polynomials chosen to retain the x$\backslash$deg term $-$ justify?

$($c$)$ How should a CRC polynomial be designed to intercept all errors occurring in an odd number of bit positions? $($Hint: Assume that $($s $+1)$ is a factor of G$($s$),$ the divisor polynomial, and then consider the divisibility of B $($x$)$ characterizing this situation, by G$($a$)).$

$($d$)$ Two CSMA$/$CD stations are each trying to transmit long $($multi$-$frame$)$ files. After each frame is sent, they contend for the channel, using the binary exponential back$-$off algorithm. What is the probability that the contention ends on round k$,$ and what is the mean number of rounds per contention period?