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

CONCEPT: A degree-r CRC generator is represented by an (r + 1)-bit divisor. The sender appends r zero bits, performs modulo-2 division, and replaces those zeros with the r-bit remainder.

A valid codeword has zero remainder when the entire transmitted frame is divided by the same generator, because the appended remainder cancels the message remainder.

1. APPLICATION: The polynomial x4 + x + 1 has coefficients 1, 0, 0, 1, 1, so its binary divisor is 10011 and r = 4.

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

3. Apply XOR with 10011 at each leading 1. The working states are:

   1. Start: 11010110110000.

   2. After the first aligned XOR: 01001110110000.

   3. After the next aligned XOR: 00000010110000.

   4. After the third aligned XOR: 00000000101000.

   5. After the final aligned XOR: 00000000001110; the last four bits give remainder 1110.

4. Replace the appended zeros with 1110: 1101011011 + 1110 = 11010110111110.

CROSS-CHECK: Dividing 11010110111110 by 10011 gives syndrome 0000. Dividing the nearby candidate 11010110111011 by the same generator gives syndrome 0101, so the zero-syndrome check distinguishes the constructed frame.

Result: the transmitted frame is 11010110111110.