$(6,3)$ systematic linear block code encodes the information sequence x $=($x$1,$ x$2,$ x$3)$

into codeword c $=($c $1,$ c $2,$ c $3,$ c $4,$ c $5,$ c $6),$ such that c $4$ is a parity check on c $1$ and c $2,$ to

make the overall parity even $($i$.$e$.,$ Ci EB c $2$ EB c$4=0).$ Similarly c $5$ is a parity check on c $2$

and c $3,$ and c$6$ is a parity check on c $1$ and c $3.$

$1.$ Determine the generator matrix of this code.

$2.$ Find the parity check matrix for this code.

$3.$ Using the parity check matrix, determine the minimum distance of this code.

$4.$ How many errors is this code capable of correcting?

$5.$ If the received sequence $($using hard decision decoding$)$ is y $=100000,$ what is the

transmitted sequence using a maximum$-$likelihood decoder? $($Assume that the crossover

probability of the channel is less than $!.)$