A computer network uses polynomials over $GF(2)$ for error checking with 8 bits as information bits and uses $x^3 + x + 1$ as the generator polynomial to generate the check bits. In this network, the message 01011011 is transmitted as

To determine the transmitted codeword using Cyclic Redundancy Check (CRC):

1. Generator Polynomial:

   $$G(x) = x^3 + x + 1 = 1 \cdot x^3 + 0 \cdot x^2 + 1 \cdot x^1 + 1 \cdot x^0$$

   The generator bit string is 1011, and the degree of the polynomial is $r = 3$.

2. Message with Appended Zeros: The original 8-bit message is 01011011. Appending $r = 3$ zeros yields:

   $$\text{Dividend} = 01011011000$$

3. Modulo-2 Binary Division:

   • Discard leading zeros: 1011011000

   • 1011011000 XOR (1011 shifted by 6) = 0000011000 = 11000

   • 11000 XOR (1011 shifted by 1) = 01110 = 1110

   • 1110 XOR 1011 = 101

   The 3-bit remainder (CRC check bits) is 101.

4. Transmitted Codeword:

   $$\text{Transmitted Message} = \text{Message} \parallel \text{Remainder} = 01011011101$$

Hence, the correct option is C.