Discussion
What is the single error-correcting code for the information code 10111 for odd parity ?
Answer: A
The code word is 100111110.
Five data bits need four Hamming check bits (\(2^{4} \ge 5 + 4 + 1\)), so the code has 9 bits. Number the positions 9 down to 1 from left to right; check bits sit at positions 1, 2, 4 and 8 and the data 1 0 1 1 1 fills positions 9, 7, 6, 5 and 3.
For odd parity, each check bit makes the count of 1s in its group odd:
\[ \begin{aligned} P_{1} &: \text{bits } 3, 5, 7, 9 = 1, 1, 0, 1 \Rightarrow P_{1} = 0 \\ P_{2} &: \text{bits } 3, 6, 7 = 1, 1, 0 \Rightarrow P_{2} = 1 \\ P_{4} &: \text{bits } 5, 6, 7 = 1, 1, 0 \Rightarrow P_{4} = 1 \\ P_{8} &: \text{bit } 9 = 1 \Rightarrow P_{8} = 0 \end{aligned} \]
Reading positions 9 to 1 gives 1 0 0 1 1 1 1 1 0. Options (C) and (D) have only 8 bits, too few for single-error correction of 5 data bits, and 111000001 does not contain the data bits at the data positions.
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