CRC for data 1010011110 with divisor 1011

Given: data = 1010011110 (10 bits), divisor (generator) = 1011 (4 bits, i.e. x³ + x + 1).

Step 1: append zeros

The divisor has 4 bits, so append 3 zeros (divisor length − 1):

Dividend = 1010011110 + 000 = 1010011110000 (13 bits)

Step 2: modulo-2 (XOR) division

Step	Working bits	Action	XOR result	Remainder (3 bits)
1	1010	1010 ⊕ 1011	0001	001
2	0010 (bring down 0)	leading 0, ⊕ 0000	0010	010
3	0101 (bring down 1)	leading 0, ⊕ 0000	0101	101
4	1011 (bring down 1)	1011 ⊕ 1011	0000	000
5	0001 (bring down 1)	leading 0	0001	001
6	0011 (bring down 1)	leading 0	0011	011
7	0110 (bring down 0)	leading 0	0110	110
8	1100 (bring down 0)	1100 ⊕ 1011	0111	111
9	1110 (bring down 0)	1110 ⊕ 1011	0101	101
10	1010 (bring down 0)	1010 ⊕ 1011	0001	001

Remainder = 001, so the CRC = 001.

Step 3: transmitted codeword

Data + CRC = 1010011110 001 = 1010011110001

Step 4: check at the receiver

Divide the received word 1010011110001 by 1011. The steps are identical up to step 9 (remainder 101). In step 10 the last bit is now 1:

• Step 10: bring down 1 → 1011 ⊕ 1011 = 0000 → remainder 000

The remainder is zero, so no error is detected and the answer is verified.

(Quotient = 1001000111. If a bit were flipped in transit the remainder would be non-zero.)