Added 6-sublot BER surface plot; Added pgf externalization

latex/presentations/midterm/presentation.bib

@@ -25,7 +25,7 @@
     journal = {IEEE Transactions on Information Theory},
     author  = {Berlekamp, E. and McEliece, R. and van Tilborg, H.},
     year    = {1978},
-    month   = {May},
+    month   = {5},
     pages   = {384–386},
 %    url     = {https://authors.library.caltech.edu/5607/1/BERieeetit78.pdf}
 }
@@ -64,7 +64,7 @@
     url = {https://doi.org/10.1561/2400000003},
     doi = {10.1561/2400000003},
     journal = {Found. Trends Optim.},
-    month = {jan},
+    month = {1},
     pages = {127–239},
     numpages = {113}
 }

latex/presentations/midterm/presentation.tex

@@ -23,6 +23,8 @@
 %\setbeamertemplate{note page}[plain]
 %\setbeameroption{show notes on second screen=right}
 \usepgfplotslibrary{colorbrewer}
+\usetikzlibrary{external}
+\tikzexternalize[prefix=build/]
 
 \usepackage{csquotes}
 \usepackage[citestyle=numeric, style=alphabetic, backend=biber,

latex/presentations/midterm/sections/decoding_algorithms.tex

@@ -83,7 +83,7 @@
 \begin{frame}[t, fragile]
     \frametitle{Proximal Decoding: Algorithm}
     \begin{itemize}
-        \item Resulting terative decoding algorithm:
+        \item Resulting terative decoding algorithm \cite{proximal_paper}:
             \vspace{2mm}
             \begin{algorithm}[caption={}, label={}]
 $\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$

latex/presentations/midterm/sections/examination_results.tex

@@ -95,7 +95,9 @@
                                      legend style={at={(0.05,0.05)},anchor=south west},
                                      ymin=3e-7, ymax=1.5,]
                     \foreach \gamma in {0.01, 0.05, 0.15}{
-                        \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{\gamma}] {res/2d_ber_fer_dfr_20433484_proximal.csv};
+                        \addplot table [x=SNR, y=BER,
+                                        col sep=comma, discard if not={gamma}{\gamma}]
+                                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
                         \legend{\gamma}
                     }
                     \legend{$\gamma=0.01$, $\gamma=0.05$, $\gamma=0.15$}
@@ -111,13 +113,26 @@
                              xlabel={SNR},
                              ylabel={$\gamma$},
                              zlabel={BER},]
-                    \addplot3[surf, mesh/rows=17, mesh/cols=14, colormap/viridis] table [col sep=comma, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv};
+                    \addplot3[surf,
+                              mesh/rows=17, mesh/cols=14,
+                              colormap/viridis] table [col sep=comma,
+                                                       x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
                     \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
-                    \addplot3[red, line width=1.5] table[col sep=comma, discard if not={gamma}{0.05}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv};
+                    \addplot3[red, line width=1.5] table [col sep=comma,
+                                                          discard if not={gamma}{0.05},
+                                                          x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
                     \addlegendentry{$\gamma = 0.05$}
-                    \addplot3[blue, line width=1.5] table[col sep=comma, discard if not={gamma}{0.01}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv};
+                    \addplot3[blue, line width=1.5] table [col sep=comma,
+                                                           discard if not={gamma}{0.01},
+                                                           x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
                     \addlegendentry{$\gamma = 0.01$}
-                    \addplot3[brown, line width=1.5] table[col sep=comma, discard if not={gamma}{0.15}, x=SNR, y=gamma, z=BER] {res/2d_ber_fer_dfr_20433484_proximal.csv};
+                    \addplot3[brown, line width=1.5] table [col sep=comma,
+                                                            discard if not={gamma}{0.15},
+                                                            x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
                     \addlegendentry{$\gamma = 0.15$}
                 \end{axis}
             \end{tikzpicture}
@@ -134,6 +149,215 @@
 \end{frame}
 
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[t]
+    \frametitle{Proximal Decoding: Choice of $\gamma$}
+   
+    \begin{figure}[H]
+        \vspace*{-0.5cm}
+        \centering
+        \begin{subfigure}[c]{0.33\textwidth}
+            \centering
+            \begin{tikzpicture}[scale=0.35]
+                \begin{axis}[view={75}{60},
+                             zmode=log,
+                             xlabel={SNR},
+                             ylabel={$\gamma$},
+                             zlabel={BER},]
+                    \addplot3[surf,
+                              mesh/rows=17, mesh/cols=10,
+                              colormap/viridis] table [col sep=comma,
+                                                       x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_963965.csv};
+                    \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
+                    \addplot3[red, line width=1.5] table[col sep=comma,
+                                                         discard if not={gamma}{0.05},
+                                                         x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_963965.csv};
+                    \addlegendentry{$\gamma = 0.05$}
+                    \addplot3[blue, line width=1.5] table[col sep=comma,
+                                                          discard if not={gamma}{0.01},
+                                                          x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_963965.csv};
+                    \addlegendentry{$\gamma = 0.01$}
+                    \addplot3[brown, line width=1.5] table[col sep=comma,
+                                                           discard if not={gamma}{0.15},
+                                                           x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_963965.csv};
+                    \addlegendentry{$\gamma = 0.15$}
+                \end{axis}
+            \end{tikzpicture}
+            \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=96, k=48$\\ \cite[Code: 96.3.965]{mackay_enc}}
+        \end{subfigure}%
+        \begin{subfigure}[c]{0.33\textwidth}
+            \centering
+            \begin{tikzpicture}[scale=0.35]
+                \begin{axis}[view={75}{60},
+                             zmode=log,
+                             xlabel={SNR},
+                             ylabel={$\gamma$},
+                             zlabel={BER},]
+                    \addplot3[surf,
+                              mesh/rows=17, mesh/cols=14,
+                              colormap/viridis] table [col sep=comma,
+                                                       x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20433484.csv};
+                    \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
+                    \addplot3[red, line width=1.5] table[col sep=comma,
+                                                         discard if not={gamma}{0.05},
+                                                         x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20433484.csv};
+                    \addlegendentry{$\gamma = 0.05$}
+                    \addplot3[blue, line width=1.5] table[col sep=comma,
+                                                          discard if not={gamma}{0.01},
+                                                          x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20433484.csv};
+                    \addlegendentry{$\gamma = 0.01$}
+                    \addplot3[brown, line width=1.5] table[col sep=comma,
+                                                           discard if not={gamma}{0.15},
+                                                           x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20433484.csv};
+                    \addlegendentry{$\gamma = 0.15$}
+                \end{axis}
+            \end{tikzpicture}
+            \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=204, k=102$\\ \cite[Code: 204.33.484]{mackay_enc}}
+        \end{subfigure}%
+        \begin{subfigure}[c]{0.33\textwidth}
+            \centering
+            \begin{tikzpicture}[scale=0.35]
+                \begin{axis}[view={75}{60},
+                             zmode=log,
+                             xlabel={SNR},
+                             ylabel={$\gamma$},
+                             zlabel={BER},]
+                    \addplot3[surf,
+                              mesh/rows=17, mesh/cols=10,
+                              colormap/viridis] table [col sep=comma,
+                                                       x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_40833844.csv};
+                    \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
+                    \addplot3[red, line width=1.5] table[col sep=comma,
+                                                         discard if not={gamma}{0.05},
+                                                         x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_40833844.csv};
+                    \addlegendentry{$\gamma = 0.05$}
+                    \addplot3[blue, line width=1.5] table[col sep=comma,
+                                                          discard if not={gamma}{0.01},
+                                                          x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_40833844.csv};
+                    \addlegendentry{$\gamma = 0.01$}
+                    \addplot3[brown, line width=1.5] table[col sep=comma,
+                                                           discard if not={gamma}{0.15},
+                                                           x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_40833844.csv};
+                    \addlegendentry{$\gamma = 0.15$}
+                \end{axis}
+            \end{tikzpicture}
+            \caption{$\left( 3, 6 \right)$-regular LDPC code with $n=408, k=204$\\ \cite[Code: 408.33.844]{mackay_enc}}
+        \end{subfigure}
+        \begin{subfigure}[c]{0.33\textwidth}
+            \centering
+            \begin{tikzpicture}[scale=0.35]
+                \begin{axis}[view={75}{60},
+                             zmode=log,
+                             xlabel={SNR},
+                             ylabel={$\gamma$},
+                             zlabel={BER},]
+                    \addplot3[surf,
+                              mesh/rows=17, mesh/cols=10,
+                              colormap/viridis] table [col sep=comma,
+                                                       x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_bch_31_26.csv};
+                    \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
+                    \addplot3[red, line width=1.5] table[col sep=comma,
+                                                         discard if not={gamma}{0.05},
+                                                         x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_bch_31_26.csv};
+                    \addlegendentry{$\gamma = 0.05$}
+                    \addplot3[blue, line width=1.5] table[col sep=comma,
+                                                          discard if not={gamma}{0.01},
+                                                          x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_bch_31_26.csv};
+                    \addlegendentry{$\gamma = 0.01$}
+                    \addplot3[brown, line width=1.5] table[col sep=comma,
+                                                           discard if not={gamma}{0.15},
+                                                           x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_bch_31_26.csv};
+                    \addlegendentry{$\gamma = 0.15$}
+                \end{axis}
+            \end{tikzpicture}
+            \caption{BCH code with $n=31, k=26$}
+        \end{subfigure}%
+        \begin{subfigure}[c]{0.33\textwidth}
+            \centering
+            \begin{tikzpicture}[scale=0.35]
+                \begin{axis}[view={75}{60},
+                             zmode=log,
+                             xlabel={SNR},
+                             ylabel={$\gamma$},
+                             zlabel={BER},]
+                    \addplot3[surf,
+                              mesh/rows=17, mesh/cols=10,
+                              colormap/viridis] table [col sep=comma,
+                                                       x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20455187.csv};
+                    \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
+                    \addplot3[red, line width=1.5] table[col sep=comma,
+                                                         discard if not={gamma}{0.05},
+                                                         x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20455187.csv};
+                    \addlegendentry{$\gamma = 0.05$}
+                    \addplot3[blue, line width=1.5] table[col sep=comma,
+                                                          discard if not={gamma}{0.01},
+                                                          x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20455187.csv};
+                    \addlegendentry{$\gamma = 0.01$}
+                    \addplot3[brown, line width=1.5] table[col sep=comma,
+                                                           discard if not={gamma}{0.15},
+                                                           x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_20455187.csv};
+                    \addlegendentry{$\gamma = 0.15$}
+                \end{axis}
+            \end{tikzpicture}
+            \caption{$\left( 5, 10 \right)$-regular LDPC code with $n=204, k=102$\\ \cite[Code: 204.55.187]{mackay_enc}}
+        \end{subfigure}%
+        \begin{subfigure}[c]{0.33\textwidth}
+            \centering
+            \begin{tikzpicture}[scale=0.35]
+                \begin{axis}[view={75}{60},
+                             zmode=log,
+                             xlabel={SNR},
+                             ylabel={$\gamma$},
+                             zlabel={BER},]
+                    \addplot3[surf,
+                              mesh/rows=17, mesh/cols=10,
+                              colormap/viridis] table [col sep=comma,
+                                                       x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_pegreg252x504.csv};
+                    \addlegendentry{$\gamma = \left[ 0\text{:}.01\text{:}.16 \right] $}
+                    \addplot3[red, line width=1.5] table[col sep=comma,
+                                                         discard if not={gamma}{0.05},
+                                                         x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_pegreg252x504.csv};
+                    \addlegendentry{$\gamma = 0.05$}
+                    \addplot3[blue, line width=1.5] table[col sep=comma,
+                                                          discard if not={gamma}{0.01},
+                                                          x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_pegreg252x504.csv};
+                    \addlegendentry{$\gamma = 0.01$}
+                    \addplot3[brown, line width=1.5] table[col sep=comma,
+                                                           discard if not={gamma}{0.15},
+                                                           x=SNR, y=gamma, z=BER]
+                                                    {res/2d_ber_fer_dfr_pegreg252x504.csv};
+                    \addlegendentry{$\gamma = 0.15$}
+                \end{axis}
+            \end{tikzpicture}
+            \caption{LDPC code (Progressive Edge Growth Construction) with $n=504, k=252$\\ \cite[Code: PEGReg252x504]{mackay_enc}}
+        \end{subfigure}%
+    \end{figure}
+\end{frame}
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[t, fragile]
     \frametitle{Proximal Decoding: Frame Error Rate}
@@ -149,7 +373,10 @@
     \begin{minipage}{.4\textwidth}
         \centering
 
-        \begin{algorithm}[caption={}, label={},
+        \begin{figure}[htpb]
+            \centering
+
+            \begin{algorithm}[caption={}, label={},
             basicstyle=\fontsize{7.5}{9.5}\selectfont
             ]
 $\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
@@ -161,7 +388,10 @@ for $k=0$ to $K-1$ do
     If $\boldsymbol{\hat{x}}$ passes the parity check condition, break the loop.
 end for
 Output $\boldsymbol{\hat{x}}$
-        \end{algorithm}
+            \end{algorithm}
+
+            \caption{Proximal decoding algorithm \cite{proximal_paper}}
+        \end{figure}
     \end{minipage}%
     \begin{minipage}{.6\textwidth}
         \centering
@@ -234,17 +464,118 @@ Output $\boldsymbol{\hat{x}}$
 \end{frame}
 
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\newcommand{\tikzbracemark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
+%\newcommand{\tikzbracemark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
+%
+%\newcommand*{\AddNote}[4]{%
+%    \begin{tikzpicture}[overlay, remember picture]
+%        \draw [decoration={brace,amplitude=0.5em},decorate,ultra thick]
+%            ($(#3)!([yshift=1.5ex]#1)!($(#3)-(0,1)$)$) --  
+%            ($(#3)!(#2)!($(#3)-(0,1)$)$)
+%                node [align=center, text width=2cm, pos=0.5, anchor=west] {#4};
+%    \end{tikzpicture}
+%}%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\begin{frame}[t, fragile]
+%    \frametitle{Proximal Decoding: Improvement using ``ML-on-List''}
+%    \setcounter{footnote}{0} 
+%
+%    \begin{itemize}
+%        \item Comparison of proximal \& hybrid-proximal-ML\\
+%            decoding simulation
+%            \footnote{(3,6) regular LDPC Code with $n=204, k=102$
+%                \cite[Code: 204.33.484]{mackay_enc}}
+%            results
+%    \end{itemize}
+%
+%    \begin{minipage}{.4\textwidth}
+%        \centering
+%
+%        \begin{algorithm}[caption={}, label={},
+%            basicstyle=\fontsize{6.5}{7.5}\selectfont
+%            ]
+%$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$$\hspace{4.185cm}\tikzbracemark{prox-start}$
+%for $k=0$ to $K-1$ do
+%    $\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
+%    Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
+%    $\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
+%    $\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
+%    If $\boldsymbol{\hat{x}}$ passes the parity check condition, output $\boldsymbol{\hat{x}}$
+%end for $\tikzbracemark{prox-end}$
+%Find $N$ most probably wrong bits $\hspace{2cm}\tikzbracemark{ml-start}$
+%Generate variations $\boldsymbol{\tilde{x}}_n$ of $\boldsymbol{\hat{x}}$ with the $N$ bits modified
+%Compute $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right] $
+%Output $\boldsymbol{\tilde{x}}_n$ with lowest $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)$ $\tikzbracemark{ml-end}$
+%        \end{algorithm}
+%
+%    \AddNote{prox-start}{prox-end}{prox-start}{\small Proximal\\Decoding}
+%    \AddNote{ml-start}{ml-end}{ml-start}{\small ML-on-List}
+%    \end{minipage}%
+%    \begin{minipage}{.6\textwidth}
+%        \centering
+%        \begin{figure}[H]
+%            \centering
+%            \vspace*{-12mm}
+%        
+%            \begin{tikzpicture}[scale=0.42]
+%                \begin{axis}[
+%                    grid=both,
+%                    xlabel={SNR}, ylabel={BER},
+%                    ymode=log,
+%                    legend style={at={(0.05,0.05)},anchor=south west},
+%                    ymax=1.5, ymin=3e-8,
+%                ]
+%
+%                \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
+%                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
+%                \addlegendentry{proximal}
+%                \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
+%                    {res/2d_ber_fer_dfr_20433484_hybrid.csv};
+%                \addlegendentry{hybrid prox. \& ML}
+%                \end{axis}
+%            \end{tikzpicture}
+%            \begin{tikzpicture}[scale=0.42]
+%                \begin{axis}[
+%                    grid=both,
+%                    xlabel={SNR}, ylabel={FER},
+%                    ymode=log,
+%                    legend style={at={(0.05,0.05)},anchor=south west},
+%                    ymax=1.5, ymin=3e-8,
+%                ]
+%
+%                \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
+%                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
+%                \addlegendentry{proximal}
+%                \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
+%                    {res/2d_ber_fer_dfr_20433484_hybrid.csv};
+%                \addlegendentry{hybrid prox. \& ML}
+%                \end{axis}
+%            \end{tikzpicture}\\
+%            \begin{tikzpicture}[scale=0.42]
+%                \begin{axis}[
+%                    grid=both,
+%                    xlabel={SNR}, ylabel={Decoding Failure Rate},
+%                    ymode=log,
+%                    legend style={at={(0.05,0.05)},anchor=south west},
+%                    ymax=1.5, ymin=3e-8,
+%                ]
+%
+%                \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
+%                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
+%                \addlegendentry{proximal}
+%                \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
+%                    {res/2d_ber_fer_dfr_20433484_hybrid.csv};
+%                \addlegendentry{hybrid prox. \& ML}
+%                \end{axis}
+%            \end{tikzpicture}
+%        
+%            \caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200, N=12$}
+%            \label{fig:simulation_results_hybrid}
+%        \end{figure}
+%    \end{minipage} 
+%\end{frame}
 
-\newcommand*{\AddNote}[4]{%
-    \begin{tikzpicture}[overlay, remember picture]
-        \draw [decoration={brace,amplitude=0.5em},decorate,ultra thick]
-            ($(#3)!([yshift=1.5ex]#1)!($(#3)-(0,1)$)$) --  
-            ($(#3)!(#2)!($(#3)-(0,1)$)$)
-                node [align=center, text width=2cm, pos=0.5, anchor=west] {#4};
-    \end{tikzpicture}
-}%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[t, fragile]
@@ -252,96 +583,56 @@ Output $\boldsymbol{\hat{x}}$
     \setcounter{footnote}{0} 
 
     \begin{itemize}
-        \item Comparison of proximal \& hybrid-proximal-ML\\
+        \item Improvement of proximal decoding by adding an ``ML-on-list'' step after iterating
-            decoding simulation
-            \footnote{(3,6) regular LDPC Code with $n=204, k=102$
-                \cite[Code: 204.33.484]{mackay_enc}}
-            results
     \end{itemize}
 
-    \begin{minipage}{.4\textwidth}
+    \begin{minipage}[t]{.48\textwidth}
         \centering
 
-        \begin{algorithm}[caption={}, label={},
+        \begin{figure}
-            basicstyle=\fontsize{6.5}{7.5}\selectfont
+           \centering
+
+            \begin{algorithm}[caption={}, label={},
+            basicstyle=\fontsize{7.5}{9.5}\selectfont
             ]
-$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$$\hspace{4.185cm}\tikzbracemark{prox-start}$
+$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
 for $k=0$ to $K-1$ do
     $\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
     Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
     $\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
     $\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
-    If $\boldsymbol{\hat{x}}$ passes the parity check condition, output $\boldsymbol{\hat{x}}$
+    If $\boldsymbol{\hat{x}}$ passes the parity check condition, break the loop.
-end for $\tikzbracemark{prox-end}$
+end for
-Find $N$ most probably wrong bits $\hspace{2cm}\tikzbracemark{ml-start}$
+Output $\boldsymbol{\hat{x}}$
-Generate variations $\boldsymbol{\tilde{x}}_n$ of $\boldsymbol{\hat{x}}$ with the $N$ bits modified
+            \end{algorithm}
-Compute $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right] $
+
-Output $\boldsymbol{\tilde{x}}_n$ with lowest $d\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)$ $\tikzbracemark{ml-end}$
+            \caption{Proximal decoding algorithm \cite{proximal_paper}}
-        \end{algorithm}
+        \end{figure}
 
-    \AddNote{prox-start}{prox-end}{prox-start}{\small Proximal\\Decoding}
-    \AddNote{ml-start}{ml-end}{ml-start}{\small ML-on-List}
     \end{minipage}%
-    \begin{minipage}{.6\textwidth}
+    \hfill\begin{minipage}[t]{.48\textwidth}
         \centering
-        \begin{figure}[H]
+        \begin{figure}
-            \centering
+           \centering 
-            \vspace*{-12mm}
-        
-            \begin{tikzpicture}[scale=0.42]
-                \begin{axis}[
-                    grid=both,
-                    xlabel={SNR}, ylabel={BER},
-                    ymode=log,
-                    legend style={at={(0.05,0.05)},anchor=south west},
-                    ymax=1.5, ymin=3e-8,
-                ]
 
-                \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
+            \begin{algorithm}[caption={}, label={},
-                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
+            basicstyle=\fontsize{7.5}{9.5}\selectfont
-                \addlegendentry{proximal}
+            ]
-                \addplot table [x=SNR, y=BER, col sep=comma, discard if not={gamma}{0.05}]
+$\boldsymbol{s}^{\left( 0 \right)} = \boldsymbol{0}$
-                    {res/2d_ber_fer_dfr_20433484_hybrid.csv};
+for $k=0$ to $K-1$ do
-                \addlegendentry{hybrid prox. \& ML}
+    $\boldsymbol{r}^{\left( k+1 \right)} = \boldsymbol{s}^{(k)} - \omega \nabla L \left( \boldsymbol{s}^{(k)}; \boldsymbol{y} \right) $
-                \end{axis}
+    Compute $\nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right)$
-            \end{tikzpicture}
+    $\boldsymbol{s}^{\left( k+1 \right)} = \boldsymbol{r}^{(k+1)} - \gamma \nabla h\left( \boldsymbol{r}^{\left( k+1 \right) } \right) $
-            \begin{tikzpicture}[scale=0.42]
+    $\boldsymbol{\hat{x}} = \text{sign}\left( \boldsymbol{s}^{\left( k+1 \right) } \right) $
-                \begin{axis}[
+    $\textcolor{KITblue}{\text{If }\boldsymbol{\hat{x}}\text{ passes the parity check condition, output }\boldsymbol{\hat{x}}}$
-                    grid=both,
+end for
-                    xlabel={SNR}, ylabel={FER},
+$\textcolor{KITblue}{\text{Find }N\text{ most probably wrong bits.}}$
-                    ymode=log,
+$\textcolor{KITblue}{\text{Generate variations } \boldsymbol{\tilde{x}}_n\text{ of }\boldsymbol{\hat{x}}\text{ with the }N\text{ bits modified.}}$
-                    legend style={at={(0.05,0.05)},anchor=south west},
+$\textcolor{KITblue}{\text{Compute }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right]}$
-                    ymax=1.5, ymin=3e-8,
+$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)}$
-                ]
+            \end{algorithm}
 
-                \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
+            \caption{Hybrid proximal \& ML decoding algorithm}
-                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
-                \addlegendentry{proximal}
-                \addplot table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                    {res/2d_ber_fer_dfr_20433484_hybrid.csv};
-                \addlegendentry{hybrid prox. \& ML}
-                \end{axis}
-            \end{tikzpicture}\\
-            \begin{tikzpicture}[scale=0.42]
-                \begin{axis}[
-                    grid=both,
-                    xlabel={SNR}, ylabel={Decoding Failure Rate},
-                    ymode=log,
-                    legend style={at={(0.05,0.05)},anchor=south west},
-                    ymax=1.5, ymin=3e-8,
-                ]
-
-                \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
-                    {res/2d_ber_fer_dfr_20433484_proximal.csv};
-                \addlegendentry{proximal}
-                \addplot table [x=SNR, y=DFR, col sep=comma, discard if not={gamma}{0.05}]
-                    {res/2d_ber_fer_dfr_20433484_hybrid.csv};
-                \addlegendentry{hybrid prox. \& ML}
-                \end{axis}
-            \end{tikzpicture}
-        
-            \caption{Simulation results for $\gamma = 0.05, \omega = 0.05, K=200, N=12$}
-            \label{fig:simulation_results_hybrid}
         \end{figure}
     \end{minipage} 
 \end{frame}
@@ -376,7 +667,7 @@ end for
 Output $\boldsymbol{\hat{x}}$
             \end{algorithm}
 
-            \caption{Proximal decoding algorithm}
+            \caption{Proximal decoding algorithm \cite{proximal_paper}}
         \end{figure}
 
     \end{minipage}%
@@ -398,8 +689,8 @@ for $k=0$ to $K-1$ do
 end for
 $\textcolor{KITblue}{\text{Find }N\text{ most probably wrong bits.}}$
 $\textcolor{KITblue}{\text{Generate variations } \boldsymbol{\tilde{x}}_n\text{ of }\boldsymbol{\hat{x}}\text{ with the }N\text{ bits modified.}}$
-$\textcolor{KITblue}{\text{Compute }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right) \forall n \in \left[ 1 : N-1 \right]}$
+$\textcolor{KITblue}{\text{Compute }\langle \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \rangle \forall n \in \left[ 1 : N-1 \right]}$
-$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }d_H\left( \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \right)}$
+$\textcolor{KITblue}{\text{Output }\boldsymbol{\tilde{x}}_n\text{ with lowest }\langle \boldsymbol{ \tilde{x}}_n, \boldsymbol{\hat{x}} \rangle}$
             \end{algorithm}
 
             \caption{Hybrid proximal \& ML decoding algorithm}