Determine all the code words of the code if Generator matrix for a (7,4) block code is as given below. Also if transmitted code word C=1011is received as 1111, use syndrome to correct this error.1 & 0 & 0 & 0 & : & 0 & 1 & 1 \\0 & 1 & 0 & 0 & : & 1 & 0 & 1 \\0 & 0 & 1 & 0 & : & 1 & 1 & 0 \\0 & 0 & 0 & 1 & : & 1 & 1 & 1 \\\end{bmatrix} \]

To determine all the code words of the code, we need to find all possible combinations of the message bits multiplied by the generator matrix.

The generator matrix given is:

$G = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

To find the code words, we multiply each possible combination of the message bits (4 bits in total) by the generator matrix.

The code words can be obtained by multiplying the message bits with the generator matrix using matrix multiplication.

For example, to find the first code word, we multiply the message bits [0 0 0 0] by the generator matrix:

$[0 0 0 0] \times \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = [0 0 0 0]$

Similarly, we can find the other code words by multiplying the remaining possible combinations of the message bits by the generator matrix.

To correct the error in the received code word C=1011, we need to calculate the syndrome. The syndrome is obtained by multiplying the received code word by the transpose of the generator matrix.

The received code word is [1 1 1 1], and the transpose of the generator matrix is:

$G^T = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

To calculate the syndrome, we multiply the received code word by the transpose of the generator matrix:

$[1 1 1 1] \times \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = [1 1 1 1]$

Since the syndrome is not zero, it indicates that an error has occurred. To correct the error, we need to find the error pattern. The error pattern is obtained by multiplying the syndrome by the inverse of the generator matrix.

The inverse of the generator matrix is the same as the generator matrix itself:

$G^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

To find the error pattern, we multiply the syndrome by the inverse of the generator matrix:

$[1 1 1 1] \times \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = [1 1 1 1]$

The error pattern indicates that all the bits in the received code word are in error. To correct the error, we need to flip the bits in the received code word.

Therefore, the corrected code word is [0 0 0 0].