Consider an $(n,k)$ block code with the following generator matrix. Find the following:

$($a$)$ The values of $n$ and $k.$

$($b$)$ The detection and correction capacities of the code.

$($c$)$ The codeword $c$ for the message $m=1001.$

$($d$)$ The syndrome for the received vector $r=1101011.$

$G=\left[\begin{array}{ccccccc}0& 1& 0& 1& 1& 0& 0\\ 1& 0& 1& 0& 1& 0& 0\\ 0& 1& 1& 0& 0& 1& 0\\ 1& 1& 0& 0& 0& 0& 1\end{array}\right]$

$($e$)$ Assume this code is used on a channel with $P_{be}=0\mathrm{.}01.$ What is the probability of receiving codeword free from error?