\appendix

\section{Supplementary Slides}%
\label{app:Supplementary Slides}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
    \frametitle{Comparison: Decoding Performance}
    
    \captionsetup[subfigure]{font=footnotesize}

    \hspace*{-0.4cm}
    \begin{minipage}[c]{0.9\textwidth}
        \centering
    
        \begin{figure}[H]
            \vspace*{-0.8cm}
            \centering
        
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                        ymode=log,
                        ymax=1.5, ymin=8e-5,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]

                        \addplot[scol2, line width=1pt, solid]
                            table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                                {res/proximal/2d_ber_fer_dfr_963965.csv};
                        \addplot[scol2, only marks, mark=square, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma,
                                   discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
                                {res/hybrid/2d_ber_fer_dfr_963965.csv};
                        \addplot[KITorange, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                                %{res/hybrid/2d_ber_fer_dfr_963965.csv};
                                {res/admm/ber_2d_963965.csv};
                        \addplot[Black, line width=1pt]
                            table [col sep=comma, x=SNR, y=FER,]
                                {res/generic/fer_ml_9633965.csv};
                    \end{axis}
                \end{tikzpicture}

                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
                    \citereference{Mac23, 96.33.965}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                        ymode=log,
                        ymax=1.5, ymin=8e-5,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                
                        \addplot[scol2, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma,
                                   discard if not={gamma}{0.05},
                                   discard if gt={SNR}{5.5}]
                                {res/proximal/2d_ber_fer_dfr_20433484.csv};
                        \addplot[scol2, only marks, mark=square, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma,
                                   discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
                                {res/hybrid/2d_ber_fer_dfr_20433484.csv};
                        \addplot[KITorange, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma,
                                   discard if not={mu}{3.0},
                                   discard if gt={SNR}{5.5}]
                                {res/admm/ber_2d_20433484.csv};
                        \addplot[Black, line width=1pt]
                            table [col sep=comma, x=SNR, y=FER,
                                   discard if gt={SNR}{5.5}]
                                {res/generic/fer_ml_20433484.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
                    \citereference{Mac23, 204.33.484}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                        ymode=log,
                        ymax=1.5, ymin=8e-5,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                
                        \addplot[scol2, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                                {res/proximal/2d_ber_fer_dfr_40833844.csv};
                        \addplot[scol2, only marks, mark=square, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma,
                                   discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
                                {res/hybrid/2d_ber_fer_dfr_40833844.csv};
                        \addplot[KITorange, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                                {res/admm/ber_2d_40833844.csv};
                    \end{axis}
                \end{tikzpicture}

                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$
                    \citereference{Mac23, 408.33.844}}
            \end{subfigure}%

            \vspace*{-2mm}

            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                        ymode=log,
                        ymax=1.5, ymin=8e-5,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                
                        \addplot[scol2, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                                {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
                        \addplot[scol2, only marks, mark=square, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma,
                                   discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
                                {res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
                        \addplot[KITorange, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                                {res/admm/ber_2d_bch_31_26.csv};
                        \addplot[Black, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma,
                                   discard if gt={SNR}{5.5},
                                   discard if lt={SNR}{1},
                                   ]
                                {res/generic/fer_ml_bch_31_26.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{BCH code with $n=31, k=26$}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                        ymode=log,
                        ymax=1.5, ymin=8e-5,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                
                        \addplot[scol2, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                                {res/proximal/2d_ber_fer_dfr_20455187.csv};
                        \addplot[scol2, only marks, mark=square, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                                {res/hybrid/2d_ber_fer_dfr_20455187.csv};
                        \addplot[KITorange, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                                {res/admm/ber_2d_20455187.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
                    \citereference{Mac23, 204.55.187}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                        ymode=log,
                        ymax=1.5, ymin=8e-5,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                
                        \addplot[scol2, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                                {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
                        \addplot[scol2, only marks, mark=square, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma,
                                   discard if not={gamma}{0.05}, discard if gt={SNR}{5.5}]
                                {res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
                        \addplot[KITorange, line width=1pt]
                            table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                                {res/admm/ber_2d_pegreg252x504.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
                
                \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
                    \citereference{Mac23, PEGReg252x504}}
            \end{subfigure}%
        \end{figure}
    \end{minipage}%
    \begin{minipage}[c]{0.1\textwidth}
        \centering
    
        \begin{figure}
            \centering
            \vspace*{-1.2cm}
            \hspace*{2mm}
            \begin{tikzpicture}[scale=0.8]
                \begin{axis}[hide axis,
                             xmin=10, xmax=50,
                             ymin=0, ymax=0.4,
                             legend columns=1,
                             legend style={draw=white!15!black}]
                    \addlegendimage{scol2, line width=1pt, solid}
                    \addlegendentry{Proximal}

                        \addlegendimage{scol2, line width=1pt, only marks, mark=square}
                    \addlegendentry{Improved}
                   
                    \addlegendimage{KITorange, line width=1pt}
                    \addlegendentry{ADMM}

                    \addlegendimage{Black, line width=1pt}
                    \addlegendentry{ML}
                \end{axis}
            \end{tikzpicture}
        \end{figure}
    \end{minipage}

    \vspace*{-1mm}

    \addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
        Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t, fragile]
    \frametitle{Comparison: Algorithm Structure}

    \vspace*{-1cm}

    \begin{figure}[h]
    \centering
   
    \begin{subfigure}[b]{0.47\textwidth}
        \centering
   
        \begin{align*}
            \text{minimize}\hspace{2mm}   & \textcolor{KITblue}{\underbrace{L\left(
                    \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right)}_{\text{Likelihood}}}
                    + \textcolor{KITred}{\underbrace{\gamma h\left( \tilde{\boldsymbol{x}}
                        \right)} _{\text{Constraints}}} \\
            \text{subject to}\hspace{2mm} &\tilde{\boldsymbol{x}} \in \mathbb{R}^n
        \end{align*}
       
        \vspace*{-5mm}

        \begin{algorithm}[caption={}, label={},
            basicstyle=\fontsize{10}{18}\selectfont
            ]
Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
while stopping critierion unfulfilled do
    $\textcolor{KITblue}{\boldsymbol{r} \leftarrow \boldsymbol{r}
        + \omega \nabla L\left( \boldsymbol{y} \mid \boldsymbol{s} \right)} $
    $\textcolor{KITred}{\boldsymbol{s} \leftarrow
        \textbf{prox}_{\gamma h}\left( \boldsymbol{r} \right)} $|\Suppressnumber|
|\Reactivatenumber|
end while
return $\hat{\boldsymbol{c}}$
        \end{algorithm}
    \end{subfigure}\hfill%
    \begin{subfigure}[b]{0.5\textwidth}
        \centering
        
        \begin{align*}
            \text{minimize}\hspace{5mm} &
                \textcolor{KITblue}{\underbrace{\boldsymbol{\gamma}^\text{T}
                    \tilde{\boldsymbol{c}}}_{\text{Likelihood}}}
                + \textcolor{KITred}{\underbrace{\sum\nolimits_{j\in\mathcal{J}}
                    g_j\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
                    \right)}_{\text{Constraints}}} \\
            \text{subject to}\hspace{5mm} &
                \tilde{\boldsymbol{c}} \in \left[ 0, 1 \right]^n
        \end{align*}
        
        \vspace*{-5mm}
    
        \begin{algorithm}[caption={}, label={},
            basicstyle=\fontsize{10}{18}\selectfont
            ]
Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}, \boldsymbol{u}, \boldsymbol{\gamma}, \rho$
while stopping criterion unfulfilled do
    $\textcolor{KITblue}{\tilde{\boldsymbol{c}} \leftarrow \argmin_{\tilde{\boldsymbol{c}}}
        \left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
        + \frac{\rho}{2}\sum_{j\in\mathcal{J}} \left\Vert
            \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j
            + \boldsymbol{u}_j \right\Vert \right)}$
    $\textcolor{KITred}{\boldsymbol{z}_j \leftarrow \textbf{prox}_{g_j}
        \left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
        + \boldsymbol{u}_j \right), \hspace{5mm}\forall j\in\mathcal{J}}$
    $\boldsymbol{u}_j \leftarrow \boldsymbol{u}_j
        + \tilde{\boldsymbol{c}} - \boldsymbol{z}_j, \hspace{14.5mm}\forall j\in\mathcal{J}$
end while
return $\hat{\boldsymbol{c}}$
        \end{algorithm}
    \end{subfigure}%
\end{figure}%

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
    \frametitle{Comparison: Time Complexity}

    \vspace*{-6.5mm}

    \begin{itemize}
        \item Codes: BCH $\left( 31, 11 \right)$; BCH $\left( 31, 26 \right)$;
            \citereference{Mac23, 96.3.965; 204.33.484; 204.55.187; 408.33.844; PEGReg252x504}
        \item Measured performance: $\sim\SI{10000}{}$ frames/s
            on Intel Core i7-7700HQ @ 2.80GHz; $n=204$
        \item Both algorithms are $\mathcal{O}\left( n \right)$ on average
    \end{itemize}

    \bigskip

    \begin{figure}[H]
        \centering
        
        \begin{tikzpicture}
            \begin{axis}[
                grid=both,
                xlabel={$n$}, ylabel={time per frame (ms)},
                legend style={at={(0.05,0.6)},anchor=south west},
                legend cell align={left},
                width=0.43\textwidth,
                height=0.32\textwidth,
            ]
                \addplot[scol0, only marks, mark=square*]
                    table [col sep=comma, x=n, y=spf,
                           y expr=\thisrow{spf} * 1000]
                        {res/proximal/fps_vs_n.csv};
                \addlegendentry{proximal}

                \addplot[scol1, only marks, mark=*]
                    table [col sep=comma, x=n, y=spf,
                           y expr=\thisrow{spf} * 1000]
                        {res/hybrid/fps_vs_n.csv};
                \addlegendentry{improved ($N=12$)}

                \addplot[scol2, only marks, mark=triangle*]
                    table [col sep=comma, x=n, y=spf,
                           y expr=\thisrow{spf} * 1000]
                        {res/admm/fps_vs_n.csv};
                \addlegendentry{ADMM}
            \end{axis}    
        \end{tikzpicture}
    \end{figure}

    \smallskip
    \smallskip

    \addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
        Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
    \frametitle{Comparison: Convergence Behavior}

    \vspace*{-6mm}

    \begin{itemize}
        \item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
        \item Minimum number of iterations independent of SNR
    \end{itemize}

    \smallskip

    \begin{figure}[H]
        \centering
 
        \begin{subfigure}[t]{0.5\textwidth}
            \centering
        
            \begin{tikzpicture}
                \begin{axis}[
                    grid=both,
                    xlabel={Iterations},
                    ylabel={Average $\lVert \boldsymbol{c}-\boldsymbol{\hat{c}} \rVert$},
                    width=0.9\textwidth,
                    height=0.675\textwidth,
                    %legend pos=outer north east,
                ]
                \addplot [scol0, mark=none, line width=1pt]
                    table [col sep=comma,
                           discard if not={omega}{0.0774263682681127},
                           discard if gt={k}{101},
                           x=k, y=err]
                        {res/proximal/2d_avg_error_20433484_1db.csv};
                \addlegendentry{$E_b / N_0 = \SI{1}{dB}$}
                \addplot [scol1, mark=none, line width=1pt]
                    table [col sep=comma, discard if not={omega}{0.0774263682681127},
                           discard if gt={k}{101},
                           x=k, y=err]
                        {res/proximal/2d_avg_error_20433484_3db.csv};
                \addlegendentry{$E_b / N_0 = \SI{3}{dB}$}
                \addplot [scol2, mark=none, line width=1pt]
                    table [col sep=comma, discard if not={omega}{0.052233450742668434},
                           discard if gt={k}{101},
                           x=k, y=err]
                        {res/proximal/2d_avg_error_20433484_5db.csv};
                \addlegendentry{$E_b / N_0 = \SI{5}{dB}$}
                \addplot [KITorange, mark=none, line width=1pt]
                    table [col sep=comma, discard if not={omega}{0.052233450742668434},
                           discard if gt={k}{101},
                           x=k, y=err]
                        {res/proximal/2d_avg_error_20433484_8db.csv};
                \addlegendentry{$E_b / N_0 = \SI{8}{dB}$}
                \end{axis}
            \end{tikzpicture}
            \caption{Proximal Decoding}
        \end{subfigure}%
        \begin{subfigure}[t]{0.5\textwidth}
            \centering
        
            \begin{tikzpicture}
                \begin{axis}[
                    grid=both,
                    width=0.9\textwidth,
                    height=0.675\textwidth,
%                    legend pos=outer north east,
                    xlabel={Iteration},
                    ylabel={Average $\left\Vert \hat{\boldsymbol{c}} - \boldsymbol{c} \right\Vert$}
                ]
                    \addplot[scol0, line width=1pt]
                        table [col sep=comma, x=k, y=err,
                               discard if not={SNR}{1.0},
                               discard if gt={k}{101}]
                            {res/admm/avg_error_20433484.csv};
                    \addlegendentry{$E_b / N_0 = \SI{1}{dB}$}

                    \addplot[scol1, line width=1pt]
                        table [col sep=comma, x=k, y=err,
                               discard if not={SNR}{2.0},
                               discard if gt={k}{101}]
                            {res/admm/avg_error_20433484.csv};
                    \addlegendentry{$E_b / N_0 = \SI{2}{dB}$}

                    \addplot[scol2, line width=1pt]
                        table [col sep=comma, x=k, y=err,
                               discard if not={SNR}{3.0},
                               discard if gt={k}{101}]
                            {res/admm/avg_error_20433484.csv};
                    \addlegendentry{$E_b / N_0 = \SI{3}{dB}$}

                    \addplot[KITorange, line width=1pt]
                        table [col sep=comma, x=k, y=err,
                               discard if not={SNR}{4.0},
                               discard if gt={k}{101}]
                            {res/admm/avg_error_20433484.csv};
                    \addlegendentry{$E_b / N_0 = \SI{4}{dB}$}
                \end{axis}
            \end{tikzpicture}
            \caption{LP Decoding using ADMM}
        \end{subfigure}%
    \end{figure}

    \addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
        Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
    \frametitle{Proximal Decoding: Choice of $\gamma$}
    
    \captionsetup[subfigure]{font=footnotesize}

    \hspace*{-0.5cm}
    \begin{minipage}[c]{0.9\textwidth}
        \centering
    
        \begin{figure}[H]
            \vspace*{-0.7cm}
            \centering
        
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[view={75}{30},
                                 zmode=log,
                                 xlabel={$E_b / N_0$ (dB)},
                                 ylabel={$\gamma$},
                                 zlabel={BER},
                                 width=1.2\textwidth,
                                 height=0.825\textwidth,
                                 legend pos=outer north east,]

                        \addplot3[surf,
                                  mesh/rows=17, mesh/cols=10,
                                  colormap/viridis] table [col sep=comma,
                                                           x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_963965.csv};
                        \addplot3[scol1, line width=1.5] table [col sep=comma,
                                                               discard if not={gamma}{0.01},
                                                               x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_963965.csv};
                        \addplot3[scol2, line width=1.5] table [col sep=comma,
                                                              discard if not={gamma}{0.05},
                                                              x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_963965.csv};
                        \addplot3[scol0, line width=1.5] table [col sep=comma,
                                                                discard if not={gamma}{0.15},
                                                                x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_963965.csv};
                    \end{axis}
                \end{tikzpicture}

                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
                    \citereference{Mac23, 96.3.965}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[view={75}{30},
                                 zmode=log,
                                 xlabel={$E_b / N_0$ (dB)},
                                 ylabel={$\gamma$},
                                 zlabel={BER},
                                 width=1.2\textwidth,
                                 height=0.825\textwidth,
                                 legend pos=outer north east,]

                        \addplot3[surf,
                                  mesh/rows=17, mesh/cols=14,
                                  colormap/viridis] table [col sep=comma,
                                                           x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_20433484.csv};
                        \addplot3[scol1, line width=1.5] table [col sep=comma,
                                                               discard if not={gamma}{0.01},
                                                               x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_20433484.csv};
                        \addplot3[scol2, line width=1.5] table [col sep=comma,
                                                              discard if not={gamma}{0.05},
                                                              x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_20433484.csv};
                        \addplot3[scol0, line width=1.5] table [col sep=comma,
                                                                discard if not={gamma}{0.15},
                                                                x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_20433484.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
                    \citereference{Mac23, 204.33.484}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[view={75}{30},
                                 zmode=log,
                                 xlabel={$E_b / N_0$ (dB)},
                                 ylabel={$\gamma$},
                                 zlabel={BER},
                                 width=1.2\textwidth,
                                 height=0.825\textwidth,
                                 legend pos=outer north east,]

                        \addplot3[surf,
                                  mesh/rows=17, mesh/cols=10,
                                  colormap/viridis] table [col sep=comma,
                                                           x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_40833844.csv};
                        \addplot3[scol1, line width=1.5] table [col sep=comma,
                                                               discard if not={gamma}{0.01},
                                                               x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_40833844.csv};
                        \addplot3[scol2, line width=1.5] table [col sep=comma,
                                                              discard if not={gamma}{0.05},
                                                              x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_40833844.csv};
                        \addplot3[scol0, line width=1.5] table [col sep=comma,
                                                                discard if not={gamma}{0.15},
                                                                x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_40833844.csv};
                    \end{axis}
                \end{tikzpicture}

                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$
                    \citereference{Mac23, 408.33.844}}
            \end{subfigure}%

            \vspace*{-2mm}

            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[view={75}{30},
                                 zmode=log,
                                 xlabel={$E_b / N_0$ (dB)},
                                 ylabel={$\gamma$},
                                 zlabel={BER},
                                 width=1.2\textwidth,
                                 height=0.825\textwidth,
                                 legend pos=outer north east,]

                        \addplot3[surf,
                                  mesh/rows=17, mesh/cols=10,
                                  colormap/viridis] table [col sep=comma,
                                                           x=SNR, y=gamma, z=BER]
                                                    {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
                        \addplot3[scol1, line width=1.5] table [col sep=comma,
                                                               discard if not={gamma}{0.01},
                                                               x=SNR, y=gamma, z=BER]
                                                    {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
                        \addplot3[scol2, line width=1.5] table [col sep=comma,
                                                              discard if not={gamma}{0.05},
                                                              x=SNR, y=gamma, z=BER]
                                                    {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
                    \addplot3[scol0, line width=1.5] table [col sep=comma,
                                                                discard if not={gamma}{0.15},
                                                                x=SNR, y=gamma, z=BER]
                                                    {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{BCH code with $n=31, k=26$}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[view={75}{30},
                                 zmode=log,
                                 xlabel={$E_b / N_0$ (dB)},
                                 ylabel={$\gamma$},
                                 zlabel={BER},
                                 width=1.2\textwidth,
                                 height=0.825\textwidth,
                                 legend pos=outer north east,]

                        \addplot3[surf,
                                  mesh/rows=17, mesh/cols=10,
                                  colormap/viridis] table [col sep=comma,
                                                           x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_20455187.csv};
                        \addplot3[scol1, line width=1.5] table [col sep=comma,
                                                               discard if not={gamma}{0.01},
                                                               x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_20455187.csv};
                        \addplot3[scol2, line width=1.5] table [col sep=comma,
                                                              discard if not={gamma}{0.05},
                                                              x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_20455187.csv};
                        \addplot3[scol0, line width=1.5] table [col sep=comma,
                                                                discard if not={gamma}{0.15},
                                                                x=SNR, y=gamma, z=BER]
                                                        {res/proximal/2d_ber_fer_dfr_20455187.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
                    \citereference{Mac23, 204.55.187}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[view={75}{30},
                                 zmode=log,
                                 xlabel={$E_b / N_0$ (dB)},
                                 ylabel={$\gamma$},
                                 zlabel={BER},
                                 width=1.2\textwidth,
                                 height=0.825\textwidth,
                                 legend pos=outer north east,]

                        \addplot3[surf,
                                  mesh/rows=17, mesh/cols=10,
                                  colormap/viridis] table [col sep=comma,
                                                           x=SNR, y=gamma, z=BER]
                                                {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
                    \addplot3[scol1, line width=1.5] table [col sep=comma,
                                                               discard if not={gamma}{0.01},
                                                               x=SNR, y=gamma, z=BER]
                                                {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
                    \addplot3[scol2, line width=1.5] table [col sep=comma,
                                                              discard if not={gamma}{0.05},
                                                              x=SNR, y=gamma, z=BER]
                                                {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
                    \addplot3[scol0, line width=1.5] table [col sep=comma,
                                                                discard if not={gamma}{0.15},
                                                                x=SNR, y=gamma, z=BER]
                                                {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
                
                \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
                    \citereference{Mac23, PEGReg252x504}}
            \end{subfigure}%
        \end{figure}
    \end{minipage}%
    \begin{minipage}[c]{0.1\textwidth}
        \centering
    
        \begin{figure}
            \centering
            \vspace*{-2cm}
            \hspace*{-3mm}
            \begin{tikzpicture}[scale=0.8]
                \begin{axis}[hide axis,
                             xmin=10, xmax=50,
                             ymin=0, ymax=0.4,
                             legend style={draw=white!15!black,legend cell align=left}]
                    \addlegendimage{surf, colormap/viridis}
                    \addlegendentry{$\gamma = \left[ 0\text{:}0.01\text{:}0.16 \right] $};
                    \addlegendimage{scol1, line width=1.5pt}
                    \addlegendentry{$\gamma = 0.01$};
                    \addlegendimage{scol2, line width=1.5pt}
                    \addlegendentry{$\gamma = 0.05$};
                    \addlegendimage{scol0, line width=1.5pt}
                    \addlegendentry{$\gamma = 0.15$};
                \end{axis}
            \end{tikzpicture}
        \end{figure}
    \end{minipage}
    
    \vspace*{-1mm}
    
    \addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
        Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
	\frametitle{LP Decoding using ADMM: Choice of $\mu$ and $\rho$}

	\vspace*{-5mm}

	\begin{itemize}
		\item (3,6) regular LDPC code with $n=204, k=102$ \citereference{Mac23, 204.33.484}
		\item No clear optimum value
	\end{itemize}

	\bigskip

	\begin{figure}[H]
		\centering

		\begin{subfigure}[c]{0.48\textwidth}
			\centering

			\begin{tikzpicture}
				\begin{axis}[
						grid=both,
						xlabel={$\mu$}, ylabel={FER},
						ymode=log,
						width=0.8\textwidth,
						height=0.6\textwidth,
					]
					\addplot[scol0, line width=1pt]
					table [col sep=comma, x=mu, y=FER,
							discard if not={SNR}{2.0},]
						{res/admm/ber_2d_20433484.csv};
					\addplot[scol1, line width=1pt]
					table [col sep=comma, x=mu, y=FER,
							discard if not={SNR}{3.0},]
						{res/admm/ber_2d_20433484.csv};
					\addplot[scol2, line width=1pt]
					table [col sep=comma, x=mu, y=FER,
							discard if not={SNR}{4.0},]
						{res/admm/ber_2d_20433484.csv};
				\end{axis}
			\end{tikzpicture}
		\end{subfigure}%
		\begin{subfigure}[c]{0.48\textwidth}
			\centering

			\begin{tikzpicture}
				\begin{axis}[
						grid=both,
						xlabel={$\rho$}, ylabel={FER},
						ymode=log,
						width=0.8\textwidth,
						height=0.6\textwidth,
					]
					\addplot[scol0, line width=1pt]
					table [col sep=comma, x=rho, y=FER,
							discard if not={SNR}{2.0},]
						{res/admm/ber_2d_20433484_rho.csv};
					\addplot[scol1, line width=1pt]
					table [col sep=comma, x=rho, y=FER,
							discard if not={SNR}{3.0},]
						{res/admm/ber_2d_20433484_rho.csv};
					\addplot[scol2, line width=1pt]
					table [col sep=comma, x=rho, y=FER,
							discard if not={SNR}{4.0},]
						{res/admm/ber_2d_20433484_rho.csv};
				\end{axis}
			\end{tikzpicture}
		\end{subfigure}%

		\begin{subfigure}[t]{\textwidth}
			\centering

			\begin{tikzpicture}
				\begin{axis}[hide axis,
						xmin=10, xmax=50,
						ymin=0, ymax=0.4,
						legend columns=3,
						legend style={draw=white!15!black,legend cell align=left}]

					\addlegendimage{scol0, line width=1pt}
					\addlegendentry{$E_b / N_0 = \SI{2}{dB}$}
					\addlegendimage{scol1, line width=1pt}
					\addlegendentry{$E_b / N_0 = \SI{3}{dB}$}
					\addlegendimage{scol2, line width=1pt}
					\addlegendentry{$E_b / N_0 = \SI{4}{dB}$}
				\end{axis}
			\end{tikzpicture}
		\end{subfigure}
	\end{figure}%

	\bigskip

	\addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
		Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
    \frametitle{LP Decoding using ADMM: Choice of $\mu$ and $\rho$}
    
    \captionsetup[subfigure]{font=footnotesize}

    \begin{minipage}[c]{0.9\textwidth}
        \centering
        \hspace*{-4mm}
    
        \begin{figure}[H]
            \centering
            \vspace*{-7mm}
        
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$\rho$}, ylabel={Avg. \# iter.},
                        ymode=log,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                        \addplot[scol1, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{2.0},]
                                {res/admm/mu_rho_kavg_963965.csv};
                        \addplot[scol2, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{5.0},]
                                {res/admm/mu_rho_kavg_963965.csv};
                        \addplot[scol0, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{9.0},]
                                {res/admm/mu_rho_kavg_963965.csv};
                    \end{axis}
                \end{tikzpicture}

                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$
                    \citereference{Mac23, 96.3.965}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$\rho$}, ylabel={Avg. \# iter.},
                        ymode=log,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                        \addplot[scol1, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{2.0},]
                                {res/admm/mu_rho_kavg_20433484.csv};
                        \addplot[scol2, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{5.0},]
                                {res/admm/mu_rho_kavg_20433484.csv};
                        \addplot[scol0, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{9.0},]
                                {res/admm/mu_rho_kavg_20433484.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$
                    \citereference{Mac23, 204.33.484}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$\rho$}, ylabel={Avg. \# iter.},
                        ymode=log,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                        \addplot[scol1, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{2.0},]
                                {res/admm/mu_rho_kavg_40833844.csv};
                        \addplot[scol2, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{5.0},]
                                {res/admm/mu_rho_kavg_40833844.csv};
                        \addplot[scol0, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{9.0},]
                                {res/admm/mu_rho_kavg_40833844.csv};
                    \end{axis}
                \end{tikzpicture}

                \vspace*{-2mm}
            
                \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$
                    \citereference{Mac23, 408.33.844}}
            \end{subfigure}%

            \vspace*{-2mm}

            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$\rho$}, ylabel={Avg. \# iter.},
                        ymode=log,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                        \addplot[scol1, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{2.0},]
                                {res/admm/mu_rho_kavg_bch_31_26.csv};
                        \addplot[scol2, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{5.0},]
                                {res/admm/mu_rho_kavg_bch_31_26.csv};
                        \addplot[scol0, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{9.0},]
                                {res/admm/mu_rho_kavg_bch_31_26.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{BCH code with $n=31, k=26$}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$\rho$}, ylabel={Avg. \# iter.},
                        ymode=log,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                        \addplot[scol1, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{2.0},]
                                {res/admm/mu_rho_kavg_20455187.csv};
                        \addplot[scol2, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{5.0},]
                                {res/admm/mu_rho_kavg_20455187.csv};
                        \addplot[scol0, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{9.0},]
                                {res/admm/mu_rho_kavg_20455187.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
            
                \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$
                    \citereference{Mac23, 204.55.187}}
            \end{subfigure}%
            \begin{subfigure}[t]{0.33\textwidth}
                \centering
            
                \begin{tikzpicture}[scale=0.8]
                    \begin{axis}[
                        grid=both,
                        xlabel={$\rho$}, ylabel={Avg. \# iter.},
                        ymode=log,
                        width=1.2\textwidth,
                        height=0.825\textwidth,
                    ]
                        \addplot[scol1, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{2.0},]
                                {res/admm/mu_rho_kavg_pegreg252x504.csv};
                        \addplot[scol2, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{5.0},]
                                {res/admm/mu_rho_kavg_pegreg252x504.csv};
                        \addplot[scol0, line width=1pt]
                            table [col sep=comma, x=rho, y=k_avg,
                                   discard if not={mu}{9.0},]
                                {res/admm/mu_rho_kavg_pegreg252x504.csv};
                    \end{axis}
                \end{tikzpicture}
            
                \vspace*{-2mm}
                
                \caption{LDPC code (progressive edge growth construction) with $n=504, k=252$
                    \citereference{Mac23, PEGReg252x504}}
            \end{subfigure}%
        \end{figure}
    \end{minipage}%
    \begin{minipage}[c]{0.1\textwidth}
        \centering
    
        \begin{figure}
            \centering

            \vspace*{-1cm}
            \hspace*{3mm}
            \begin{tikzpicture}[scale=0.8]
                \begin{axis}[hide axis,
                             xmin=10, xmax=50,
                             ymin=0, ymax=0.4,
                             legend columns=1,
                             legend style={draw=white!15!black}]

                    \addlegendimage{scol0, line width=1pt}
                    \addlegendentry{$\mu = 9$}
                    \addlegendimage{scol2, line width=1pt}
                    \addlegendentry{$\mu = 5$}
                    \addlegendimage{scol1, line width=1pt}
                    \addlegendentry{$\mu = 2$}
                \end{axis}
            \end{tikzpicture}
        \end{figure}
    \end{minipage}
    
    \vspace*{-1mm}
    
    \addreference{Mac23}{David J.C. MacKay: \emph{Encyclopedia of Sparse Graph Codes}.
        Jan. 2023. URL: \url{http://www.inference.org.uk/mackay/codes/data.html}.}
\end{frame}