Virtual Labs

Given the linear code

\mathcal{C}

= \{(0,0,0,0,0,0),

(0,1,0,1,0,1),

(1,0,0,0,1,0),

(1,1,0,1,1,1)\}

and a codeword

\textbf{x} =

(0,1,0,1,0,1)

transmitted over the

BEC(\epsilon)

. Choose the correct option for probability of error if received vector is

\textbf{y}

= (0,1,?,1,0,?)

, where

?

represents erasures.

{\epsilon}(1-\epsilon)^5

Explanation

{\epsilon}^2(1-\epsilon)^2

Explanation

{\epsilon}^2(1-\epsilon)^4

Explanation

{\epsilon}(1-\epsilon)^4

Explanation

Consider a quaternary erasure channel

QEC(\epsilon)

which takes four input symbols,

\{0, 1, 2, 3\}

, where the channel erases the transmitted symbol with probability

\epsilon

. Choose the valid pair of transmitted and received vectors for transmission over

QEC(\epsilon)

channel.

\textbf{x} = (0,1,2,1,3) ;

\textbf{y} = (0,?,1,?,1)

Explanation

\textbf{x} = (3,3) ;

\textbf{y} = (3,1)

Explanation

\textbf{x} = (1,2,0,1,3) ;

\textbf{y} = (?,?,?,?,?)

Explanation

\textbf{x} = (0,1,0,1,1,) ;

\textbf{y} = (0,?,1,?,1)

Explanation

Consider the code

\mathcal{C}

= \{(0,0,0,0,0,0),

(1,1,0,0,1,1),

(1,1,1,0,1,1),

(0,0,1,0,0,0)\}

. Let the vector

\textbf{x}

= (1,1,1,0,1,1)

be transmitted over

BSC

. Suppose we know that

\textit{ exactly one bit }

is flipped. Which of the following options can not be a potential received vector?

(1,1,1,1,1,1)

Explanation

(0,1,1,0,1,1)

Explanation

(1,1,1,0,1,1)

Explanation

(1,1,0,0,1,1)

Explanation

Suppose we are given a code

\mathcal{C}

= \{(0,0,0),

(1,1,1)\}

. One codeword from this code,

\mathcal{C}

, is transmitted over Binary Symmetric Channel

BSC(p)

with bit-flip probability

p

. Calculate the probability of error for incorrect decoding using Maximum Likelihood Decoder.

p

Explanation

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

Explanation

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

Explanation

p^2(1-p)

Explanation

Consider the code

\mathcal{C}

=\{(0,0,0),(1,0,1),

(0,1,1),(1,1,0)\}

. Let

\textbf{x}

be a codeword (in bipolar form) is being transmitted over the AWGN channel, with mean

0

and variance

1

. Suppose the receiver receives

\textbf{y}

= (-10^3,0.5,1)

. Choose the correct Maximum Likelihood estimate for the transmitted codeword.

(0,0,0)

Explanation

(1,0,1)

Explanation

(0,1,1)

Explanation

(1,1,0)

Explanation

Maximum Likelihood Decoding of Linear Codes on Binary-Input Memoryless Channels