Linear Block Codes
1. The set of vectors
{ [ 0 1 1 1 0 ] , [ 1 1 1 1 0 ] , [ 0 1 1 1 1 ] } \left\{ \begin{bmatrix}0 \\ 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix}1 \\ 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \right\} ⎩ ⎨ ⎧ 0 1 1 1 0 , 1 1 1 1 0 , 0 1 1 1 1 ⎭ ⎬ ⎫ defined over
F 2 \mathbb{F}_2 F 2 are linearly independent. True or false?
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2. Consider the vectors
v 1 = [ 0 1 1 0 ] , v 2 = [ 1 0 1 1 ] , v 3 = [ 1 1 0 1 ] \mathbf{v}_1 = \begin{bmatrix}0 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \mathbf{v}_2 =\begin{bmatrix}1 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{v}_3 =\begin{bmatrix}1 \\ 1 \\ 0 \\ 1 \end{bmatrix} v 1 = 0 1 1 0 , v 2 = 1 0 1 1 , v 3 = 1 1 0 1 defined over
F 2 \mathbb{F}_2 F 2 . The number of vectors in span
{ v 1 , v 2 , v 3 } \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} { v 1 , v 2 , v 3 } is equal to?
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3. Consider the set
F 7 = \mathbb{F}_7 = F 7 = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } \{ 0, 1, 2, 3, 4, 5, 6 \} { 0 , 1 , 2 , 3 , 4 , 5 , 6 } . For any
a , b ∈ F 7 \mathbf{a, b} \in \mathbb{F}_7 a , b ∈ F 7 the addition and multiplication operations are defined as below. Does
F 7 \mathbb{F}_7 F 7 form a field or not?
+ 0 1 2 3 4 5 6 1 1 2 3 4 5 6 0 2 2 3 4 5 6 0 1 3 3 4 5 6 0 1 2 4 4 5 6 0 1 2 3 5 5 6 0 1 2 3 4 6 6 0 1 2 3 4 5 \begin{array}{c|ccccccc} + & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1 & 2 & 3 & 4 & 5 & 6 & 0 \\ 2 & 2 & 3 & 4 & 5 & 6 & 0 & 1 \\ 3 & 3 & 4 & 5 & 6 & 0 & 1 & 2 \\ 4 & 4 & 5 & 6 & 0 & 1 & 2 & 3 \\ 5 & 5 & 6 & 0 & 1 & 2 & 3 & 4 \\ 6 & 6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{array} + 1 2 3 4 5 6 0 1 2 3 4 5 6 1 2 3 4 5 6 0 2 3 4 5 6 0 1 3 4 5 6 0 1 2 4 5 6 0 1 2 3 5 6 0 1 2 3 4 6 0 1 2 3 4 5 ⋅ 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3 0 3 6 2 5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1 6 4 2 6 0 6 5 4 3 2 1 \begin{array}{c|ccccccc} \cdot & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 0 & 2 & 4 & 6 & 1 & 3 & 5 \\ 3 & 0 & 3 & 6 & 2 & 5 & 1 & 4 \\ 4 & 0 & 4 & 1 & 5 & 2 & 6 & 3 \\ 5 & 0 & 5 & 3 & 1 & 6 & 4 & 2 \\ 6 & 0 & 6 & 5 & 4 & 3 & 2 & 1 \\ \hline \end{array} ⋅ 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3 0 3 6 2 5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1 6 4 2 6 0 6 5 4 3 2 1
4. Consider the vectors
v 1 = [ 1 0 0 1 0 ] , v 2 = [ 1 0 0 1 1 ] , v 3 = [ 0 1 1 0 0 ] v 4 = [ 0 0 0 0 1 ] \mathbf{v}_1 = \begin{bmatrix}1 \\ 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \mathbf{v}_2 =\begin{bmatrix}1 \\ 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix}0 \\ 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} \mathbf{v}_4 = \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} v 1 = 1 0 0 1 0 , v 2 = 1 0 0 1 1 , v 3 = 0 1 1 0 0 v 4 = 0 0 0 0 1 defined over
F 2 \mathbb{F}_2 F 2 . Can you find scalars
a 1 , a 2 , a 3 , a 4 \mathbf{a_1, a_2, a_3,a_4} a 1 , a 2 , a 3 , a 4 ∈ \in ∈ F 2 \mathbb{F}_2 F 2 such that
a 1 v 1 + a 2 v 2 + a 3 v 3 + a 4 v 4 = 0 \mathbf{a_1v_1}+\mathbf{a_2v_2}+\mathbf{a_3v_3}+\mathbf{a_4v_4} = \mathbf{0} a 1 v 1 + a 2 v 2 + a 3 v 3 + a 4 v 4 = 0 ? Comment whether the set of vectors
v 1 , v 2 , v 3 , \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, v 1 , v 2 , v 3 , and
v 4 \mathbf{v}_4 v 4 are linearly independent or not.
a 1 = 0 , a 2 = 0 , a 3 = 0 , a 4 = 0 a_1=0, a_2=0, a_3=0, a_4=0 a 1 = 0 , a 2 = 0 , a 3 = 0 , a 4 = 0
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a 1 = 1 , a 2 = 1 , a 3 = 0 , a 4 = − 1 a_1=1, a_2=1, a_3=0, a_4=-1 a 1 = 1 , a 2 = 1 , a 3 = 0 , a 4 = − 1
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a 1 = 1 , a 2 = 1 , a 3 = 0 , a 4 = 1 a_1=1, a_2=1, a_3=0, a_4=1 a 1 = 1 , a 2 = 1 , a 3 = 0 , a 4 = 1
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Explanation
a 1 = 1 , a 2 = 1 , a 3 = 0 , a 4 = 1 a_1=1, a_2=1, a_3=0, a_4=1 a 1 = 1 , a 2 = 1 , a 3 = 0 , a 4 = 1
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5. The set of vectors
{ 10101 , 10010 , 01110 , 11111 , 11000 } \mathbf{\{1 0 1 0 1, 1 0 0 1 0, 0 1 1 1 0, 1 1 1 1 1, 1 1 0 0 0 \}} { 10101 , 10010 , 01110 , 11111 , 11000 } correspond to linear or non-linear block code?
6. The encoded the message sequence 0 1 0 using REP-7 code is given by
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7. The encoded the message sequence 1 0 0 0 1 0 1 1 0 1 1 1 using SPC(
4 , 3 \mathbf{4,3} 4 , 3 ) code is given by
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