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Q5(a) · Hamming error detection
2024-02-06 · AI-only Sem V variant
Exact historical variant. Current-syllabus questions solved; excluded topics and source limitations stated.
Q5(a). The seven-bit received Hamming word is 1011001. Under even parity, determine correctness and locate any erroneous bit. [10 marks]
Answer
Given
Received word 1011001; even parity. Number positions 7,6,5,4,3,2,1 from left to right, so position 1 is the rightmost bit.
To find
Syndrome, erroneous position, and corrected word.
Formula and why
LaTeX: s_p=\bigoplus_{i\,\&\,p\ne0}b_i,\quad \text{error position}=4s_4+2s_2+s_1Each parity check covers positions whose binary position number contains that parity bit. For even parity, XOR must be zero; a 1 identifies a failed group.
Solution
Step 1. Label every received bit
| Position | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| Received bit | 1 | 0 | 1 | 1 | 0 | 0 | 1 |
Step 2. Evaluate the three parity groups
| Parity | Covered positions | Bits added | Sum | Sum mod 2 |
|---|---|---|---|---|
| 1 | 1, 3, 5, 7 | 1 + 0 + 1 + 1 | 3 | 1 |
| 2 | 2, 3, 6, 7 | 0 + 0 + 0 + 1 | 1 | 1 |
| 4 | 4, 5, 6, 7 | 1 + 1 + 0 + 1 | 3 | 1 |
Step 3. Read the syndrome as a binary position
Syndrome s4s2s1 = 111; decimal = 7. Under the single-bit-error assumption, flip position 7 from the right. Corrected word = 0011001.
Step 4. State the assumption
Ordinary Hamming(7,4) corrects one bit. A multi-bit corruption may produce a misleading nonzero syndrome, so do not claim guaranteed correction of arbitrary errors.