Official PYQ · 2021Set 1 · Q29Data Link Layer: Framing, Error Detection, MAC, Ethernet BridgingMCQ2 marks

GATE CS 2021 Set 1 Q29 — Data Link Layer: Framing, Error Detection, MAC, Ethernet Bridging

Assume that a $12$ -bit Hamming codeword consisting of $8$ -bit data and $4$ check bits is $d_8d_7d_6d_5c_8d_4d_ 4d_3d_2c_4d_1c_2c_1$ , where the data bits and the check bits are given in the following tables: $\overset{\textbf{Data bits}}{\begin{array}{|c|c|c|c|} \hline d_{8} & d_{7} & d_{6} & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} \\ \hline 1 & 1 & 0 & x & 0 & 1 & 0 & 1 \\ \hline \end{array}}\qquad \overset{\textbf{Check bits}}{\begin{array}{|c|c|c|c|} \hline c_{8} & c_{4} & c_{2} & c_{1} \\ \hline y & 0 & 1 & 0 \\ \hline \end{array}}$ Which one of the following choices gives the correct values of $x$ and $y$ ?

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Answer: A

In a standard 12-bit Hamming code with even parity, the bits are indexed from 1 to 12 (right-to-left):

Position	12	11	10	9	8	7	6	5	4	3	2	1
Bit	$d_8$	$d_7$	$d_6$	$d_5$	$c_8$	$d_4$	$d_3$	$d_2$	$c_4$	$d_1$	$c_2$	$c_1$
Value	1	1	0	$x$	$y$	0	1	0	0	1	1	0

Using even parity, we set up the parity equations for the check bits:

For $c_1$ (covers positions 1, 3, 5, 7, 9, 11): $c_1 \oplus d_1 \oplus d_2 \oplus d_4 \oplus d_5 \oplus d_7 = 0$ Substituting the given values: $0 \oplus 1 \oplus 0 \oplus 0 \oplus x \oplus 1 = 0 \implies x = 0$
For $c_8$ (covers positions 8, 9, 10, 11, 12): $c_8 \oplus d_5 \oplus d_6 \oplus d_7 \oplus d_8 = 0$ Substituting the values (with $x = 0$ ): $y \oplus 0 \oplus 0 \oplus 1 \oplus 1 = 0 \implies y = 0$

Thus, $x = 0$ and $y = 0$ , which corresponds to option A.

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Published 10 Oct 2026

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