Three CRC check bits for the generator polynomial x³ + x + 1 (bit string 1011) are to be appended to the data 1101. The check bits are the remainder of dividing 1101000, the data followed by three 0s, by 1011, using exclusive OR instead of subtraction in the division. Which of the following are the check bits?

Answer: A. 001

Divide 1101000 by 1011. The exclusive OR of the leading 1101 and 1011 is 0110; bringing down the next bit gives 1100, whose exclusive OR with 1011 is 0111; 1110 with 1011 gives 0101; and finally 1010 with 1011 gives 0001, so the remainder is 001. The word sent is 1101001, and the receiver judges that there is no error if the remainder after dividing by 1011 is 0.

• A. Correct. The remainder of the division using exclusive OR is 001, and the codeword sent is 1101001.
• B. 101 is the remainder 5 obtained by dividing 1101000 and 1011 as ordinary binary numbers, 104 and 11. A CRC uses division by exclusive OR, without borrows.
• C. 110 is the remainder of dividing 1101 by 1011 without appending three 0s. The division is done after appending as many 0s as there are check bits.