1. Consider the linear block code
C ( 7 , 2 ) \mathcal{C}(7,2) C ( 7 , 2 ) generated by the following generator matrix
G \mathbf{G} G . Is this code systematic?
G = [ 1 0 0 1 1 0 1 0 1 0 0 1 1 1 ] \begin{align*} G = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 & 1 \end{bmatrix} \end{align*} G = [ 1 0 0 1 0 0 1 0 1 1 0 1 1 1 ]
2. Consider the linear block code defined by the following parity check matrix
H \mathbf{H} H . Is
v = [ 1 1 1 1 1 1 ] \mathbf{v} = \begin{bmatrix} 1 & 1 & 1& 1 & 1 & 1\end{bmatrix} v = [ 1 1 1 1 1 1 ] a codeword in this code?
H = [ 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*} H = \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*} H = 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 with the following parity check matrix
H \mathbf{H} H . What is the dimension of this code?
H = [ 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*} H = \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*} H = 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
4. Find the minimum distance of the code with the following parity check matrix.
H = [ 1 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 0 ] \begin{align*} H = \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*} H = 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
5. Consider the linear block code generated by the following generator matrix
G \mathbf{G} G . What will be the codeword corresponding to the message
[ 0 1 0 1 ] \begin{bmatrix} 0 & 1 & 0 & 1\end{bmatrix} [ 0 1 0 1 ] ?
G = [ 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 ] \begin{align*} G = \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0\\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \end{bmatrix} \end{align*} G = 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 1 1 0 1 0 0 0
Explanation
Explanation
Explanation
Explanation
Consider the following block codes. Identify the linear block codes.
C 1 = { [ 0 0 0 0 ] , [ 1 0 0 1 ] , [ 0 1 1 1 ] , [ 1 1 1 0 ] } C 2 = { [ 0 0 0 0 ] } C 3 = { [ 0 0 0 ] , [ 1 0 0 ] , [ 0 0 1 ] , [ 1 0 0 ] , [ 0 1 1 ] , [ 1 1 0 ] , [ 1 0 1 ] , [ 1 1 1 ] } \begin{align*} \mathcal{C}_1 &= \left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix} \right\} \\ \mathcal{C}_2 &= \left\{\begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\right\} \\ \mathcal{C}_3 &= \left\{\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix},\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix},\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix},\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\right\} \end{align*} C 1 C 2 C 3 = ⎩ ⎨ ⎧ 0 0 0 0 , 1 0 0 1 , 0 1 1 1 , 1 1 1 0 ⎭ ⎬ ⎫ = ⎩ ⎨ ⎧ 0 0 0 0 ⎭ ⎬ ⎫ = ⎩ ⎨ ⎧ 0 0 0 , 1 0 0 , 0 0 1 , 1 0 0 , 0 1 1 , 1 1 0 , 1 0 1 , 1 1 1 ⎭ ⎬ ⎫
Explanation
Explanation
Explanation
Explanation
