\appendix

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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[NavyBlue, 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[RedOrange, 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[ForestGreen, 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[NavyBlue, 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[RedOrange, 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[ForestGreen, 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[NavyBlue, 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[RedOrange, 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[ForestGreen, 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[NavyBlue, 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[RedOrange, 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[ForestGreen, 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[NavyBlue, 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[RedOrange, 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[ForestGreen, 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[NavyBlue, 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[RedOrange, 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[ForestGreen, 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] $};
                    \addlegendimage{NavyBlue, line width=1.5pt}
                    \addlegendentry{$\gamma = 0.01$};
                    \addlegendimage{RedOrange, line width=1.5pt}
                    \addlegendentry{$\gamma = 0.05$};
                    \addlegendimage{ForestGreen, line width=1.5pt}
                    \addlegendentry{$\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 $\rho$}
    
    \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[ForestGreen, 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[RedOrange, 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[NavyBlue, 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[ForestGreen, 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[RedOrange, 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[NavyBlue, 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[ForestGreen, 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[RedOrange, 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[NavyBlue, 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[ForestGreen, 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[RedOrange, 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[NavyBlue, 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[ForestGreen, 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[RedOrange, 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[NavyBlue, 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[ForestGreen, 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[RedOrange, 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[NavyBlue, 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{NavyBlue, line width=1pt}
                    \addlegendentry{$\mu = 9$}
                    \addlegendimage{RedOrange, line width=1pt}
                    \addlegendentry{$\mu = 5$}
                    \addlegendimage{ForestGreen, 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}

