$(2)[20$ points$]$ A Low$-$Density Parity$-$Check $($LDPC$)$ Code is defined, based on the LDPC matrix, $\backslash ($ H $\backslash )$ as following.

$\backslash [$

H$=\backslash$left$[\backslash$begin$\{$array$\}\{$lllll$\}$

$1$ & $0$ & $1$ & $0$ & $1\backslash \backslash$

$1$ & $1$ & $0$ & $1$ & $0\backslash \backslash$

$1$ & $0$ & $0$ & $1$ & $1$

$\backslash$end$\{$array$\}\backslash$right$]$

$\backslash ]$

$($a$)$ Draw a Tanner graph corresponding to the LDPC matrix $\backslash ($ H $\backslash ).[5$ points$]$

$($b$)$ The received codeword, caused by the transmission of a codeword at the transmitter and the noise at the receiver is$,\backslash (\backslash$mathbf$\{$r$\}=\backslash$left$[\backslash$begin$\{$array$\}\{$lllll$\}1$ & $0$ & $1$ & $1$ & $1\backslash$end$\{$array$\}\backslash$right$]\backslash ).$

Perform only ONE iteration of LDPC hard$-$decision decoding and describe the decoding process as in detail as possible for you to earn the full credit. Describe each and every bit sent to a check node, each and every bit returned to a variable node, and updated bit at the variable node for each and every check and variable node. $[10$ points$]$

$($c$)$ Using the given LDPC matrix, $\backslash ($ H $\backslash ),$ perform the parity check for the estimated codeword after ONE iteration in $($b$).$ Explain whether the estimated codeword in $($b$)$ can be regarded as correct codeword or errored codeword. $[5$ points$]$