Question

Assume that a 12-bit Hamming codeword consisting of 8-bit data and 4 check bits is $d_8 d_7 d_6 d_5 c_8 d_4 d_3 d_2 c_4 d_1 c_2 c_1$, where the data bits and the check bits are given in the following tables. Which one of the following choices gives the correct values of $x$ and $y$? 

• A. $x$ is 0 and $y$ is 0
• B. $x$ is 0 and $y$ is 1
• C. $x$ is 1 and $y$ is 0
• D. $x$ is 1 and $y$ is 1

Hint:

For Hamming codes, always write parity equations explicitly to avoid mistakes in determining unknown data or check bits.

Answer 1

The Correct Option is A

Step 1: Recall the structure of a (12, 8) Hamming code.
In a (12, 8) Hamming code, the parity (check) bits are placed at positions that are powers of two: 1, 2, 4, and 8.

Each check bit enforces even parity over a predefined set of bit positions that include both data bits and other check bits.

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Step 2: Determine the value of x.
The unknown bit x appears in the parity equations corresponding to the relevant check bits that cover position d₅.

Substituting the given data-bit values and enforcing the even parity condition, the parity equation is satisfied only when:

x = 0

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Step 3: Determine the value of y.
The unknown bit y is involved in the parity check for c₈, which covers positions including d₈, d₇, d₆, and d₅.

Using the known bit values along with x = 0, and again applying the even parity rule, the parity condition is satisfied only when:

y = 0

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Final Conclusion:
Both parity conditions are satisfied when:

x = 0 and y = 0