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Let H= $\begin{bmatrix} 1&0&0 \\ 1&1&0 \\ 0&1&1 \\ 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}$ be parity check matrix.

Decode the following words relative to maximum likelyhood decoding function.

1. 011001
2. 101011
3. 111010
4. 110110
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1.The syndrome of the given code word 011001 is given by

$$011001 * \begin{bmatrix} 1&0&0 \\ 1&1&0 \\ 0&1&1 \\ 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}=100$$

Since 100 is the syndrome of the code word 100110, therefore the given code 011001 is corrected as $100110 \oplus 011001= 111111$ and is decoded as 11.

2.The syndrome of the given code word 101011 is given by

$$101011 *\begin{bmatrix} 1&0&0 \\ 1&1&0 \\ 0&1&1 \\ 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}=100$$

Since 100 is the syndrome of the code word 010100, therefore the given code 101011 is corrected as $010100 \oplus 101011 = 111111$ and is decoded as 11.

3.The syndrome of the given code word 111010 is given by

$$111010 *\begin{bmatrix} 1&0&0 \\ 1&1&0 \\ 0&1&1 \\ 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}=001$$

Since 100 is the syndrome of the code word 000101, therefore the given code 111010 is corrected as $000101 \oplus 111010 = 111111$ and is decoded as 11.

4.The syndrome of the given code word 110110 is given by

$$110110 *\begin{bmatrix} 1&0&0 \\ 1&1&0 \\ 0&1&1 \\ 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}=000$$

Since 100 is the syndrome of the code word 001001, therefore the given code 110110 is corrected as $001001 \oplus 110110 = 111111$ and is decoded as 11.