A binary block code has code vectors in systematic form as given:

$v=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 1& 1& 1& 0& 0\\ 1& 0& 1& 0& 1& 0\\ 1& 1& 0& 1& 1& 0\\ 1& 1& 0& 0& 0& 1\\ 1& 0& 1& 1& 0& 1\\ 0& 1& 1& 0& 1& 1\\ 0& 0& 0& 1& 1& 1\end{array}\right]$

a$)$ What is the rate of this code?

b$)$ Write G & H of this code in systematic form?

c$)$ What is the min hamming distance of this code?

d$)$ How many errors can be detected and errors can be corrected?

e$)$ Compute the syndrome $S$ of the received vector $r=[101011].$