CRC for message 101001101101 with G(x) = x⁵ + x⁴ + x² + 1

Message: 1010011011 01 = 101001101101 (12 bits)

Generator: x⁵ + x⁴ + x² + 1 → coefficients of x⁵ x⁴ x³ x² x¹ x⁰ = 1 1 0 1 0 1 → 110101 (6 bits, degree 5)

Step 1: append 5 zeros (degree of G)

Dividend = 101001101101 + 00000 = 10100110110100000 (17 bits)

Step 2: modulo-2 division by 110101

Step	Working bits	XOR with	Result	Remainder (5 bits)
1	101001	110101	011100	11100
2	111001 (bring 1)	110101	001100	01100
3	011000 (bring 0)	000000	011000	11000
4	110001 (bring 1)	110101	000100	00100
5	001001 (bring 1)	000000	001001	01001
6	010010 (bring 0)	000000	010010	10010
7	100101 (bring 1)	110101	010000	10000
8	100000 (bring 0)	110101	010101	10101
9	101010 (bring 0)	110101	011111	11111
10	111110 (bring 0)	110101	001011	01011
11	010110 (bring 0)	000000	010110	10110
12	101100 (bring 0)	110101	011001	11001

Remainder = 11001, so the CRC = 11001.

Step 3: transmitted frame

Message + CRC = 101001101101 11001 = 10100110110111001

Step 4: verification at the receiver

Divide 10100110110111001 by 110101. Steps 1–7 are the same (remainder 10000), then:

Step	Working bits	XOR with	Remainder
8	100001 (bring 1)	110101	10100
9	101001 (bring 1)	110101	11100
10	111000 (bring 0)	110101	01101
11	011010 (bring 0)	000000	11010
12	110101 (bring 1)	110101	00000

The remainder is zero, so the frame is accepted as error-free.