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: Understand the Hamming code structure.
In a $(12,8)$ Hamming code, parity (check) bits are placed at positions that are powers of two: $1$, $2$, $4$, and $8$. Each check bit ensures even parity over a specific set of bit positions.

Step 2: Determine the value of $x$.
Using the parity equations for the relevant check bits that include $d_5$, and enforcing even parity, the value of $x$ is obtained as $0$.

Step 3: Determine the value of $y$.
Similarly, applying the parity condition for check bit $c_8$, which covers positions including $d_8$, $d_7$, $d_6$, and $d_5$, and substituting known values, the parity requirement gives $y = 0$.

Step 4: Conclusion.
Both unknowns satisfy the parity conditions when $x = 0$ and $y = 0$.