- 2072 Chaitra · 2+8 marks
All code words of the (6,3) code with given G
Short answer
A (6,3) linear block code maps 3-bit messages to 6-bit codewords via c=mG. The eight codewords are 000000, 001101, 010110, 011011, 100111, 101010, 110001, 111100.
What are block codes? The generator matrix for a (6,3) block code is shown below. Obtain all code words. G = [1 0 0 : 1 1 1; 0 1 0 : 1 1 0; 0 0 1 : 1 0 1]
Block codes
A block code (n, k) splits the data stream into blocks of k bits and maps each block to an n-bit codeword by adding n − k parity bits. There are 2ᵏ valid codewords out of 2ⁿ possible words; the code rate is k/n. For a linear block code, c = m·G (mod 2).
Codewords of the (6,3) code
G = [I₃ | P] with rows 100111, 010110, 001101. For m = (m1 m2 m3): c = [m1 m2 m3 | m1⊕m2⊕m3, m1⊕m2, m1⊕m3]
| Message m | Operation | Codeword | Weight |
|---|---|---|---|
| 000 | – | 000000 | 0 |
| 001 | row 3 | 001101 | 3 |
| 010 | row 2 | 010110 | 3 |
| 011 | row 2 ⊕ row 3 | 011011 | 4 |
| 100 | row 1 | 100111 | 4 |
| 101 | row 1 ⊕ row 3 | 101010 | 3 |
| 110 | row 1 ⊕ row 2 | 110001 | 3 |
| 111 | all rows | 111100 | 4 |
Example: 011 → 010110 ⊕ 001101 = 011011.
The minimum non-zero weight is 3, so d_min = 3 (detects 2 errors, corrects 1).
✓ Answer checked by Claude Sonnet 5.5
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