Virtual Labs

Error Detection and Correction, Standard Array, Syndrome Decoding

1. Decode

00111

using the following standard array.

\\

a: Decoded codeword is

11000

. Explanation

Explanation

b: Decoded codeword is

01010

. Explanation

Explanation

c: Decoded codeword is

11111

. Explanation

Explanation

d: Decoded codeword is

00111

. Explanation

Explanation

2. Consider the following standard array. What is the error correction capability of this code?

\\

a: 1 Explanation

Explanation

b: 2 Explanation

Explanation

c: 3 Explanation

Explanation

d: None of the above Explanation

Explanation

3. Decode

1001011

using the following syndrome table.

\\

a: Decoded codeword is

1111111

Explanation

b: Decoded codeword is

1101001

Explanation

c: Decoded codeword is

1001011

Explanation

d: Data insufficient Explanation

Explanation

4. Consider the linear block code

\{0000, 0011, 1100, 1111 \}

and set of standard array tables given below. Select the correct option for standard array of this code for performing optimal decoding over the BSC

(p)

with

p<1/2

\\

a: Standard array 1 is correct. Explanation

Explanation

b: Standard array 2 is correct. Explanation

Explanation

c: Standard array 3 is correct. Explanation

Explanation

d: Standard arrays 2 and 3 are correct. Explanation

Explanation

5. Choose the incorrect statement.

a: Coset leaders determine the error correction capability of the code. Explanation

Explanation

b: Coset leaders determine the error detection capability of the code. Explanation

Explanation

c: Let

\mathbf{e}_4

and

\mathbf{e}_5

be the coset leaders of the 4th and 5th rows of a standard array of some linear block code respectively. Then the Hamming weight of

\mathbf{e}_4

should necessarily be less than or equal to the Hamming weight of

\mathbf{e}_5

. Explanation

Explanation

d: Let

\mathbf{e}_4

and

\mathbf{e}_5

be the coset leaders of the 4th and 5th rows of a standard array of some linear block code respectively. Then the Hamming weight of

\mathbf{e}_5

can be equal to the Hamming weight of

\mathbf{e}_4

. Explanation

Explanation

6. Consider a

(7, 3)

linear block code with parity check matrix

H

. If the syndrome of received vector

\mathbf{y}

\mathbf{s} = [1, 0, 1, 1]

, and the coset leader with this syndrome is

\mathbf{e} = [0, 0, 1, 0, 0, 0, 0]

, what is the decoded codeword if

\mathbf{y} = [1, 1, 0, 1, 0, 1, 1]

[1, 1, 0, 1, 0, 1, 1]

Explanation

[1, 1, 1, 1, 0, 1, 1]

Explanation

[0, 0, 1, 0, 0, 0, 0]

Explanation

[0, 0, 0, 1, 0, 1, 1]

Explanation

7. For a

(15, 7)

linear block code, how many distinct syndromes exist?

a: 7 Explanation

Explanation

b: 8 Explanation

Explanation

c: 128 Explanation

Explanation

d: 256 Explanation

Explanation

8. In a standard array for a

(6, 3)

code, if the coset leader of row 5 is

[0, 1, 1, 0, 0, 0]

and the codeword in column 3 is

[1, 0, 1, 1, 0, 1]

, what vector is at the intersection of row 5 and column 3?

[1, 0, 1, 1, 0, 1]

Explanation

[0, 1, 1, 0, 0, 0]

Explanation

[1, 1, 0, 1, 0, 1]

Explanation

[0, 0, 0, 1, 0, 1]

Explanation

9. What is the relationship between the number of coset leaders and the dimension of the syndrome space?

a: Number of coset leaders equals the dimension Explanation

Explanation

b: Number of coset leaders is

2^{\text{dimension}}

Explanation

c: They are unrelated Explanation

Explanation

d: Number of coset leaders is twice the dimension Explanation

Explanation

10. Consider syndrome decoding for a code with parity check matrix

H

. If

\mathbf{y}_1

and

\mathbf{y}_2

are in the same coset, what is true about

\mathbf{y}_1 - \mathbf{y}_2

\mathbf{y}_1 - \mathbf{y}_2

is the all-zero vector Explanation

Explanation

\mathbf{y}_1 - \mathbf{y}_2

is a codeword Explanation

Explanation

\mathbf{y}_1 - \mathbf{y}_2

is a coset leader Explanation

Explanation

\mathbf{y}_1 - \mathbf{y}_2

has syndrome equal to the coset leader Explanation

Explanation

11. For a

(31, 26)

linear block code, what is the storage requirement ratio between standard array decoding and syndrome decoding (in terms of vectors stored)?

a: 1:1 Explanation

Explanation

2^{26}:32

Explanation

2^{31}:2^5

Explanation

2^5:2^{31}

Explanation

12. In syndrome decoding, suppose the syndrome lookup table contains an error where coset leader

\mathbf{e}_i

with syndrome

\mathbf{s}_i

is replaced by

\mathbf{e}_j

with the same syndrome. What is the consequence?

a: No effect on decoding Explanation

Explanation

b: Some received vectors will be decoded incorrectly Explanation

Explanation

c: The decoder will fail completely Explanation

Explanation

d: Only detection will be affected, not correction Explanation

Explanation

13. For a perfect code of length

n

, how are the coset leaders distributed by weight?

a: All coset leaders have weight 0 Explanation

Explanation

b: All coset leaders have weight

\leq t

where

t

is the error correction capability Explanation

Explanation

c: Coset leaders have all possible weights from 0 to

n

Explanation

d: All coset leaders have the same weight Explanation

Explanation