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Part-1: Preliminaries

Part-2: Linear Block Codes

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Aim

Theory

Pretest

Procedure

Part-1: Preliminaries

Part-2: Linear Block Codes

Posttest

References

Contributors

Feedback

Linear Block Codes

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Intermediate

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1. Consider three vectors

\mathbf{v}_1 = \begin{bmatrix} 1 & 1 & 0 & 0 & 1 \end{bmatrix}

,

\mathbf{v}_2 = \begin{bmatrix} 0 & 1 & 1 & 0 & 1 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} 0 & 0 & 1 & 0 & 1 \end{bmatrix}

defined over

\mathbb{F}_2

. What will be

\mathbf{v}_1+\mathbf{v}_2+\mathbf{v}_3?

a:

\begin{bmatrix} 1 & 2 & 2 & 0 & 3 \end{bmatrix}

Explanation

Explanation

b:

\begin{bmatrix} 1 & 0 & 1 & 0 & 1 \end{bmatrix}

Explanation

Explanation

c:

\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \end{bmatrix}

Explanation

Explanation

d:

\begin{bmatrix} 2 & 1 & 1 & 0 & 3 \end{bmatrix}

Explanation

Explanation

2. Consider a vector

\mathbf{v} = \begin{bmatrix} 1 & 1 & 0 & 0 \end{bmatrix}

and a matrix

M = \begin{bmatrix} 0 & 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ \end{bmatrix}

defined over

\mathbb{F}_2

. What will be

\mathbf{v} \cdot M

?

a:

\begin{bmatrix} 0 & 2 & 2 & 0 & 1 \end{bmatrix}

Explanation

Explanation

b:

\begin{bmatrix} 1 & 1 & 1 & 0 & 0 \end{bmatrix}

Explanation

Explanation

c:

\begin{bmatrix} 1 & 1 & 1 & 0 & 1 \end{bmatrix}

Explanation

Explanation

d:

\begin{bmatrix} 1 & 0 & 1 & 0 & 1 \end{bmatrix}

Explanation

Explanation

3. The vectors

\mathbf{v}_1 = \begin{bmatrix}1 & 1 & 0 & 1 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix}1 & 0 & 0 & 1 \end{bmatrix}

and

\mathbf{v}_3 = \begin{bmatrix} 1 & 0 & 1 & 0 \end{bmatrix}

defined over

\mathbb{F}_2

are linearly dependent. True or false?

a: True Explanation

Explanation

b: False Explanation

Explanation

c: Data insufficient Explanation

Explanation

4. The set of integers form a field. True of false?

a: True Explanation

Explanation

b: False Explanation

Explanation

5. What is the dimension of the vector space given below?

\begin{align*} U = \left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0\\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\0\\ 1\end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 0\\ 0\end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 1\end{bmatrix} \right\}. \end{align*}

a: 1 Explanation

Explanation

b: 2 Explanation

Explanation

c: 3 Explanation

Explanation

d: 4 Explanation

Explanation

6. What is the codeword of

\mathbf{(7,6)}

SPC code for the message

\mathbf{\begin{bmatrix} 0 & 1 & 1 & 0 & 1 & 1\end{bmatrix}}

?

a:

\begin{bmatrix} 0 & 1 & 1 & 0 & 1 & 1& 1\end{bmatrix}

Explanation

Explanation

b:

\begin{bmatrix} 0 & 1 & 0 & 0 & 1 & 1& 1\end{bmatrix}

Explanation

Explanation

c:

\begin{bmatrix} 0 & 1 & 1 & 1 & 1 & 1& 0\end{bmatrix}

Explanation

Explanation

d:

\begin{bmatrix} 0 & 1 & 1 & 0 & 1 & 1& 0\end{bmatrix}

Explanation

Explanation

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