Virtual Labs

1. Let

G

and

H

be generator and parity check matrices of a linear block code respectively. Then the syndrome of the vector

\mathbf{w} \in \mathbb{F}_2^n

is defined as

\mathbf{w}G^T

Explanation

\mathbf{w}H^T

Explanation

H\mathbf{w}^T

Explanation

G\mathbf{w}^T

Explanation

2. Consider the linear block code generated by the following parity matrix

H

. What is the syndrome of the vector

1001011

\begin{align*} G = \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 & 1 & 1 & 0\\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \end{bmatrix} \end{align*}

a: 011 Explanation

Explanation

b: 100 Explanation

Explanation

c: 101 Explanation

Explanation

d: None of the above Explanation

Explanation

3. Let

\mathbf{e}_2

and

\mathbf{e}_5

be the coset leaders of the 2nd and 5th rows of a standard array of some linear block code respectively. Then which of the choices of

\mathbf{e}_2

and

\mathbf{e}_5

are incorrect for an optimal decoding over BSC

(p)

with

p=0.01

\mathbf{e}_2 = 0010

and

\mathbf{e}_5=0001

Explanation

\mathbf{e}_2 = 010000

and

\mathbf{e}_5=001101

Explanation

\mathbf{e}_2 = 00110

and

\mathbf{e}_5=01000

Explanation

d: All are valid. Explanation

Explanation

4. Consider

\mathbf{y}_1, \mathbf{y}_2 \in \mathbb{F}_2^n

such that

\mathbf{y}_1 = \mathbf{v}_1 + \mathbf{e}

and

\mathbf{y}_2 = \mathbf{v}_2 + \mathbf{e}

, where

\mathbf{v}_1

and

\mathbf{v}_2

are codewords and

\mathbf{e}

is an arbitrary vector in

\mathbb{F}_2^n

. Then choose the correct statement.

\mathbf{y}_1

and

\mathbf{y}_2

have the same syndrome. Explanation

Explanation

\mathbf{y}_1

and

\mathbf{y}_2

have different syndromes. Explanation

Explanation

c: Data insufficient Explanation

Explanation

d: None of the above Explanation

Explanation

5. Standard array decoding and syndrome decoding may give different decoded codewords for the same received vector. True or false?

a: True Explanation

Explanation

b: False Explanation

Explanation

Error Detection and Correction, Standard Array, Syndrome Decoding