GATE CS 2017 Set 1 Q32 — Data Link Layer: Framing, Error Detection, MAC, Ethernet Bridging

To find the transmitted message using Cyclic Redundancy Check (CRC), we follow these steps:

1. Identify the Divisor: The generator polynomial is $G(x) = x^3 + x + 1$, which corresponds to the binary divisor $1011$.
2. Append Zeros: Since the degree of the generator polynomial is $r = 3$, we append $3$ zeros to the $8$-bit message $01011011$, giving the dividend: $$01011011000$$
3. Perform Modulo-2 Division: We divide $1011011000$ (ignoring the leading zero) by $1011$ using XOR operations:

               1000110
   1011 ) 1011011000
          1011
          0000011000  (bring down digits)
               1011
               01110
               1011
               0101   ← Remainder (3 bits)
     

   The remainder is $101$.
4. Construct the Transmitted Message: Replace the $3$ appended zeros with the remainder: $$\text{Transmitted Message} = 01011011101$$

Thus, the correct option is C.