Question $3(15$p$)$ Consider the $5$ properties of a regular LDPC code. Check the example in the lecture

notes, MacKay$-$Neal parity check matrix, for $w_c=3,$ and $w_r=4.$

a$)$ Examine the implication of $w_c=3$ and $w_r=4.$

b$)$ Give two examples for Rule #$2$ and Rule #$3$

c$)$ Give two examples for Rule #$4$ and Rule #$5$

d$)$ Does this code construction method generate symmetric code?

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

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

Check nodes $($rows of $H)$