1. Consider the linear block code generated by the following generator matrix
G \mathbf{G} G . What is the dimension of this code?
G = [ 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 ] \begin{align*} G = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \end{bmatrix} \end{align*} G = 1 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0
Explanation
Explanation
Explanation
Explanation
2. Consider the linear block code
C ( 6 , 4 ) \mathcal{C}(6,4) C ( 6 , 4 ) generated by the following generator matrix
G \mathbf{G} G . Is
v = [ 1 1 1 1 0 1 ] \mathbf{v = \begin{bmatrix} 1 & 1 & 1& 1 & 0 & 1\end{bmatrix}} v = [ 1 1 1 1 0 1 ] a codeword in this code?
G = [ 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 ] \begin{align*}G = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 0\\ 0 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 & 1 \end{bmatrix} \end{align*} G = 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1
3. Consider the linear block code generated by the following generator matrix
G G G . The generator matrix of its dual code will be
H = [ 1 1 1 1 1 ] \begin{align*} H = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \end{bmatrix} \end{align*} H = [ 1 1 1 1 1 ]
Explanation
Explanation
Explanation
Explanation
4. Consider the linear block code
C ( 6 , 3 ) \mathcal{C}(6,3) C ( 6 , 3 ) generated by the following generator matrix
G \mathbf{G} G . Is this code systematic?
G = [ 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 1 ] \begin{align*} G = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 0\\ 0 & 1 & 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 1 \end{bmatrix} \end{align*} G = 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1
5. Consider the linear block code generated by the following generator matrix
G G G . What will be the codeword corresponding to the message
[ 1 1 0 ] ? \begin{bmatrix} 1 & 1 & 0\end{bmatrix}? [ 1 1 0 ] ? G = [ 1 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 0 ] \begin{align*} G = \begin{bmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 & 1 & 1 & 1\\ 1 & 0 & 0 & 1 & 1 & 1 & 0 \end{bmatrix} \end{align*} G = 1 0 1 0 1 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0
Explanation
Explanation
Explanation
Explanation
6. Consider the matrices
G 1 , G 2 , H 1 \mathbf{G_1, G_2, H_1} G 1 , G 2 , H 1 and
H 2 \mathbf{H_2} H 2 given below. Identify generator matrix-parity check matrix pairs.
G 1 = [ 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 ] , G 2 = [ 1 0 1 0 1 1 ] H 1 = [ 1 1 1 ] , H 2 = [ 1 0 0 1 1 0 0 1 1 1 0 0 ] . \begin{align*} G_1 = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 0\\ 0 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 & 1 \end{bmatrix},\hspace{1cm} G_2 = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 1 \end{bmatrix} \\ H_1 = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}, \hspace{1cm} H_2 = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 0\\ 0 & 1 & 1 & 1 & 0 & 0 \end{bmatrix}. \end{align*} G 1 = 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 , G 2 = [ 1 0 0 1 1 1 ] H 1 = [ 1 1 1 ] , H 2 = [ 1 0 0 1 0 1 1 1 1 0 0 0 ] .
Explanation
Explanation
Explanation
Explanation
