If the data frame to be transmitted is 1101011011 and the CRC generator…

CONCEPT: In CRC, a generator polynomial of degree r is represented as an (r + 1)-bit divisor. The sender appends r zero bits to the data, performs modulo-2 division, and replaces the zeros with the r-bit remainder.

A valid transmitted frame should divide by the generator with a zero syndrome, because the appended remainder cancels the division remainder.

1. Convert the polynomial: x4 + x + 1 corresponds to the binary divisor 10011, so r = 4.

2. Append four zeros to the data: 1101011011 becomes 11010110110000.

3. Divide 11010110110000 by 10011 using XOR subtraction. The modulo-2 remainder is 1110.

4. Replace the four appended zeros with 1110, giving 1101011011 + 1110 = 11010110111110.

CROSS-CHECK: Dividing the full frame 11010110111110 by 10011 leaves syndrome 0000, while 11010110111011 leaves a nonzero syndrome.

Result: the transmitted frame is 11010110111110.