$($a$)$ All linear group block codes have an important property called $$closure$.$

Briefly describe this and demonstrate its operation with the following $(5,2)$

code:

Code Codeword

a $00000$

b $00111$

c $11100$

d $11011[5$ marks$]$

$($b$)$ A $(7,4)$ Hamming codeword is sent over a channel with a bit error rate of

$0\mathrm{.}08.$ Calculate the probability of exactly $3$ errors occurring in any codeword.

$($c$)$ Stations are allowed to transmit onto a shared radio channel whenever

they like $($i$.$e$.$ pure Aloha is in operation$).$ The transmission rate is $4$Mbps and

each station transmits fixed sized packets of $2000$ bits each. If $200$ packets are

transmitted in one second, and follow a Poisson distribution, calculate the

normalised offered load to the network $($G$)$ and from this the probability of a

collision on the radio channel.

$($d$)$ In the figure shown below, frames are generated at node A and sent to

node C through node B$.$ Determine the minimum transmission rate required

between nodes B and C so that the buffers of B are not flooded, based on the

following assumptions:

Assumptions:

$1.$ The data rate between A and B is $100$ kbps$.$

$2.$ The propagation delay is $50\backslash$mu sec$/$km for both lines.

$3.$ There are full$-$duplex lines between the nodes.

$4.$ All data frames are $1000$ bits long; ACK frames are separate frames of

negligible length.

$5.$ Between A and B$,$ a sliding window protocol with a window size of $5$ is

used. Between B and C$,$ stop$-$and$-$wait is used.

$6.$ There are no transmission errors.

Hint: In order not to flood the buffers of B$,$ the average number of frames entering

and leaving B must be the same over a long interval