GATE CS 2026 Set 2 — Question 43
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Computer Networks → Data Link Layer → Error Detection (Parity, CRC)
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Question
Consider the transmission of data bits 110001011 over a link that uses Cyclic Redundancy Check (CRC) code for error detection. If the generator bit pattern is given to be 1001, which one of the following options shows the remainder bit pattern appended to the data bits before transmission?
Correct answer
(D) 100
Solution
To find the CRC remainder, we perform binary division (modulo-2 arithmetic) of the data bits appended with zeros by the generator polynomial, where is the number of bits in the generator.Data bits (M):
Generator (G): The generator has bits. Therefore, we append zeros to the data bits.
Augmented data (M'): Now, we perform modulo-2 binary division of the augmented data by the generator. We align the generator with the leftmost '1' of the current dividend segment and perform XOR. If the current segment starts with '0', we effectively XOR with '0000' (or just bring down the next bit without XORing with the generator).
The remainder bit pattern is .Therefore, the correct option is C.
Generator (G): The generator has bits. Therefore, we append zeros to the data bits.
Augmented data (M'): Now, we perform modulo-2 binary division of the augmented data by the generator. We align the generator with the leftmost '1' of the current dividend segment and perform XOR. If the current segment starts with '0', we effectively XOR with '0000' (or just bring down the next bit without XORing with the generator).
110001011000 (Augmented Data)
^ 1001 (Generator)
-----------
01010 (Result of 1100 XOR 1001, then bring down next bit '0')
^ 1001 (Align with leftmost '1')
-----------
00110 (Result of 1010 XOR 1001, then bring down next bit '0')
^ 0000 (Current segment 0110 starts with '0', so XOR with 0000)
-----------
01101 (Result of 0110 XOR 0000, then bring down next bit '1')
^ 1001 (Align with leftmost '1')
-----------
01001 (Result of 1101 XOR 1001, then bring down next bit '1')
^ 1001 (Align with leftmost '1')
-----------
00000 (Result of 1001 XOR 1001, then bring down next bit '0')
^ 0000 (Current segment 0000 starts with '0', so XOR with 0000)
-----------
00000 (Result of 0000 XOR 0000, then bring down next bit '0')
^ 0000 (Current segment 0000 starts with '0', so XOR with 0000)
-----------
00000 (Result of 0000 XOR 0000, then bring down next bit '0')
^ 0000 (Current segment 0000 starts with '0', so XOR with 0000)
-----------
000 (Remainder - the last 3 bits, as generator is 4 bits)
The remainder bit pattern is .Therefore, the correct option is C.
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