Virtual Labs

Linear Block Codes

1. The set of vectors

\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\}

defined over

\mathbb{F}_2

are linearly independent. True or false?

a: True Explanation

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b: False Explanation

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c: Data insufficient Explanation

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2. Consider the vectors

\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}

defined over

\mathbb{F}_2

. The number of vectors in span

\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}

is equal to?

a: 1 Explanation

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b: 2 Explanation

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c: 3 Explanation

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d: 4 Explanation

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3. Consider the set

\mathbb{F}_7 =

\{ 0, 1, 2, 3, 4, 5, 6 \}

. For any

\mathbf{a, b} \in \mathbb{F}_7

the addition and multiplication operations are defined as below. Does

\mathbb{F}_7

form a field or not?

\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}

\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}

a: Yes Explanation

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b: No Explanation

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4. Consider the vectors

\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}

defined over

\mathbb{F}_2

. Can you find scalars

\mathbf{a_1, a_2, a_3,a_4}

\in

\mathbb{F}_2

such that

\mathbf{a_1v_1}+\mathbf{a_2v_2}+\mathbf{a_3v_3}+\mathbf{a_4v_4} = \mathbf{0}

? Comment whether the set of vectors

\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3,

and

\mathbf{v}_4

are linearly independent or not.

a_1=0, a_2=0, a_3=0, a_4=0

, Linearly independent vectors Explanation

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a_1=1, a_2=1, a_3=0, a_4=-1

, Linearly dependent vectors Explanation

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a_1=1, a_2=1, a_3=0, a_4=1

, Linearly independent vectors Explanation

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a_1=1, a_2=1, a_3=0, a_4=1

, Linearly dependent vectors Explanation

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5. The set of vectors

\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 \}}

correspond to linear or non-linear block code?

a: Linear block code Explanation

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b: Non-linear block code Explanation

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6. The encoded the message sequence 0 1 0 using REP-7 code is given by

a: 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 Explanation

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b: 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 Explanation

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c: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 Explanation

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d: None of the above Explanation

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7. The encoded the message sequence 1 0 0 0 1 0 1 1 0 1 1 1 using SPC(

\mathbf{4,3}

) code is given by

a: 1 0 0 0 1 1 0 1 1 1 0 1 1 1 1 Explanation

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b: 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 Explanation

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c: 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 Explanation

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d: 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 Explanation

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8. What is the Hamming weight of the vector

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

a: 3 Explanation

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b: 4 Explanation

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c: 5 Explanation

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d: 7 Explanation

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9. For the received vector

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

from a REP-5 code transmission, what is the decoded message using majority logic decoding?

a: 0 Explanation

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b: 1 Explanation

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c: Error detected, cannot decode Explanation

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d: Ambiguous Explanation

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10. Consider a

(5,3)

linear block code with the following codebook:

\mathcal{C} = \{00000, 10110, 01011, 11101, 00111, 10001, 01100, 11010\}

. What is the minimum distance of this code?

a: 1 Explanation

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b: 2 Explanation

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c: 3 Explanation

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d: 4 Explanation

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11. If the received vector from a

(5,3)

SPC code is

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

, can we detect if an error occurred?

a: Yes, an error is detected Explanation

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b: No, no error detected Explanation

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c: Cannot determine Explanation

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d: Depends on the transmitted codeword Explanation

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12. For a linear block code with parameters

(n, k, d_{min})

, what is the redundancy?

k

Explanation

n-k

Explanation

d_{min}

Explanation

n-d_{min}

Explanation

13. Consider a linear block code

\mathcal{C}(7,4,3)

. If two bits are flipped in a transmitted codeword, can the code always correct the errors?

a: Yes, always Explanation

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b: No, never Explanation

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c: Sometimes, depending on error positions Explanation

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d: Cannot be determined Explanation

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14. What is the relationship between the Hamming weight of a codeword and the Hamming distance for linear block codes?

a: Hamming weight equals Hamming distance from zero vector Explanation

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b: They are unrelated Explanation

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c: Hamming weight is always greater Explanation

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d: Hamming distance is always greater Explanation

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15. For a binary symmetric channel with crossover probability

p

, if a REP-3 codeword is transmitted and decoded using majority logic, what is the probability of decoding error?

p

Explanation

3p^2(1-p)

Explanation

3p^2(1-p) + p^3

Explanation

1 - (1-p)^3

Explanation