a$)$ A binary linear block code whose parity check matrix is shown below:

$H=\left[\begin{array}{cccccccc}0& 0& 1& 1& 1& 0& 1& 0\\ 0& 1& 0& 1& 0& 1& 1& 0\\ 1& 0& 0& 0& 1& 1& 1& 0\\ 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right]$

What is the rate of this code? What is the number of valid codewords?

b$)$ Consider a linear binary block code with the following Generator matrix:

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

i$)$ Find all the possible code vectors and develop the set of code vectors.

ii$)$ Develop the table of cosets for error detection.

iii$)$ If the received vector $r-=[001100],$ perform syndrome decoding to detect error pattern.

iv$)$ Correct error and find the detected code vector.