Consider the use of a 16 bit CRC with generator polynomial g(x) = x16 + x12 + x5 + 1. (This is known as CRC-16-CCITT, and for example is used in the Bluetooth wireless communication protocol. You might find this link for information about CRCs to be useful.)

A. Given the data polynomial d(x) in the cases below, calculate the corresponding CRC for each case (specify the CRC in polynomial form). (i): d(x) = x7 + x5 + x2 + x + 1 (ii): d(x) = x^12 + x^4 + x.

B. As discussed in class, the CRC polynomial r(x) is calculated so that the Frame polynomial F(x) = x^L d(x) + r(x) = p(x)g(x) ,where p(x) is some polynomial. The received frame is F(x) + e(x), where e(x) is the error polynomial and indicates the location of any transmission errors. For each of the cases given for the error polynomial e(x), state whether or not the errors will be properly detected by a CRC check at the receiver. (i): e(x) = x^5 , (ii): e(x) = x^21 + x^23. (iii) e(x) = x^20 + x^21 + x^22 + x^34 , (iv) e(x) = x^20 + x^21 + x^22 + x^34 + x^35, (v) e(x) = x^20 + x^21 + x^22 + x^34 + x^35 + x^36