Question $4(15$p$)$ Consider the $5$ properties of a regular LDPC code. Check the example below and answer following questions.

a$)$ What is $w_c$ and $w_r$ in this example, respectively?

b$)$ Examine whether Rule#$1$ is satisfied. That is$,$ does each code bit involve with $w_c$ parity constraints?

Further, whether or not each parity constraint $($e$.$g$.,f_1,f_2,f_3,f_4)$ involves $w_r$ bits?

c$)$ For a given parity check matrix as follows, if the received codeword is $1011-0101,$ what is the decision

at the receiver?

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

$($a$)$ Parity$-$check matrix