251 lines
10 KiB
TeX
251 lines
10 KiB
TeX
\section{Theoretical Background}%
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\label{sec:Theoretical Background}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Motivation}%
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\label{sub:Motivation}
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\begin{frame}[t]
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\frametitle{Motivation}
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\begin{itemize}
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\item The general [ML] decoding problem for linear codes and the general problem
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of finding the weights of a linear code are both NP-complete. \cite{ml_np_hard_proof}
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\item The standard message-passing algorithms used for decoding [LDPC and turbo codes]
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are often difficult to analyze. \cite{feldman_thesis}
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\item The iterative message–passing algorithms preffered in practice do not guarantee
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optimality and may fail to decode correctly when the graph contains cycles
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\cite{ldpc_conv}
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Presumptions}%
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\label{sub:Presumptions}
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\begin{frame}[t]
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\frametitle{Presumptions: Channel \& Modulation}
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\tikzstyle{mapper} = [rectangle, minimum width=1.5cm, minimum height=0.7cm,
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rounded corners=0.1cm, text centered, draw=black, fill=KITgreen!80]
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\begin{figure}[htpb]
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\centering
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\begin{tikzpicture}[scale=1, transform shape]
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\node (in) {$c\left[ k \right] $};
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\node[mapper, right=0.5cm of in] (bpskmap) {Mapper};
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\node[right=1.5cm of bpskmap,
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draw, circle, inner sep=0pt, minimum size=0.5cm] (add) {$+$};
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\node[right=0.5cm of add] (out) {$y\left[ k \right] $};
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\node[below=0.5cm of add] (noise) {$n\left[ k \right] $};
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\node at ($(bpskmap.east)!0.5!(add.west) + (0,0.3cm)$) {$x\left[ k \right] $};
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\draw[->] (in) -- (bpskmap);
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\draw[->] (bpskmap) -- (add);
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\draw[->] (add) -- (out);
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\draw[->] (noise) -- (add);
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\end{tikzpicture}
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\end{figure}
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\begin{itemize}
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\item All simulations are performed with BPSK Modulation:
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\begin{align*}
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x\left[ k \right] = \left( -1 \right)^{c\left[ k \right] },
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\hspace{5mm} \boldsymbol{c} \in \mathbb{F}_2^n,
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\hspace{2mm} k\in \left\{ 1, \ldots, n \right\}
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\end{align*}
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\item The used channel model is AWGN:
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\begin{align*}
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\boldsymbol{y} = \boldsymbol{x} + \boldsymbol{n},
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\hspace{5mm}\boldsymbol{n}\sim \mathcal{N}
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\left(0,\frac{1}{2}\left(\frac{k}{n}\frac{E_b}{N_0}\right)^{-1}\right),
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\hspace{2mm} \boldsymbol{y}, \boldsymbol{n} \in \mathbb{R}^n
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\end{align*}
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\item All-zeros assumption:
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\begin{align*}
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\boldsymbol{c} = 0
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\end{align*}
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{LP Decoding}%
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\label{sub:LP Decoding}
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\begin{frame}[t]
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\frametitle{LP Decoding}
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\begin{minipage}[c]{0.6\linewidth}
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\begin{itemize}
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\item Codeword Polytope:
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\begin{align*}
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\text{poly}\left( \mathcal{C} \right) =
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\left\{
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\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}}
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\boldsymbol{c} : \lambda_{\boldsymbol{c}} \ge 0,
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\sum_{\boldsymbol{c}\in\mathcal{C}}\lambda_{\boldsymbol{c}} = 1
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\right\},
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\hspace{5mm} \lambda_{\boldsymbol{c}} \in \mathbb{R}
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\end{align*}
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\item Cost Function:
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\begin{align*}
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\sum_{i=1}^{n} \gamma_i c_i,
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\hspace{5mm}\gamma_i = \log\left(
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\frac{P\left( Y=y_i | C=0 \right) }{P\left( Y=y_i | C=1 \right) } \right)
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\end{align*}
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\item LP Formulation of ML Decoding:
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\begin{align*}
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&\text{minimize } \sum_{i=1}^{n} \gamma_i f_i \\
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&\text{subject to } \boldsymbol{f}\in\text{poly}\left( \mathcal{C} \right)
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\end{align*}
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\end{itemize}
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\end{minipage}%
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\hfill%
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\begin{minipage}[c]{0.4\linewidth}
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\begin{figure}[H]
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{245}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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% Cube
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\draw[dashed] (0, 0, 0) -- (2, 0, 0);
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\draw[dashed] (2, 0, 0) -- (2, 0, 2);
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\draw[] (2, 0, 2) -- (0, 0, 2);
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\draw[] (0, 0, 2) -- (0, 0, 0);
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\draw[] (0, 2, 0) -- (2, 2, 0);
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\draw[] (2, 2, 0) -- (2, 2, 2);
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\draw[] (2, 2, 2) -- (0, 2, 2);
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\draw[] (0, 2, 2) -- (0, 2, 0);
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\draw[] (0, 0, 0) -- (0, 2, 0);
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\draw[dashed] (2, 0, 0) -- (2, 2, 0);
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\draw[] (2, 0, 2) -- (2, 2, 2);
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\draw[] (0, 0, 2) -- (0, 2, 2);
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% Codeword Polytope
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);
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\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
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\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);
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\draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
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% Polytope Annotations
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\node[codeword] (c000) at (0, 0, 0) {};% {$\left( 0, 0, 0 \right) $};
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\node[codeword] (c101) at (2, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
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\node[codeword] (c110) at (2, 2, 0) {};% {$\left( 1, 1, 0 \right) $};
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\node[codeword] (c011) at (0, 2, 2) {};% {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, right=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c101] {$\left( 1, 0, 1 \right) $};
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\node[color=KITblue, left=0cm of c110] {$\left( 1, 1, 0 \right) $};
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\node[color=KITblue, left=0cm of c011] {$\left( 0, 1, 1 \right) $};
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% f
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\node[color=KITgreen, fill=KITgreen,
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draw, circle, inner sep=0pt, minimum size=4pt] (f) at (0.7, 0.7, 1) {};
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\node[color=KITgreen, right=0cm of f] {$\boldsymbol{f}$};
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\end{tikzpicture}
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\caption{$\text{poly}\left( \mathcal{C} \right)$ for $n=3$}
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\end{figure}
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\end{minipage}
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\end{frame}
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\begin{frame}[t]
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\frametitle{LP Relaxation}
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\begin{minipage}[c]{0.6\linewidth}
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\begin{itemize}
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\item Set of all variable nodes incident to a check node:
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\begin{align*}
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N\left( j \right) \equiv \left\{
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i | i\in \mathcal{I},
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\boldsymbol{H}_{j,i} = 1
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\right\},
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j \in \mathcal{J}
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\end{align*}
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\begin{align*}
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S \subseteq N\left( j \right), \left| S \right| \text{odd}
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\end{align*}
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\item Relaxed polytope representation:
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\begin{align*}
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\sum_{i\in \left( N\left( j \right) \setminus S\right) } f_i
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+ \sum_{i\in S} \left( 1 - f_i \right) \ge 1
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\end{align*}
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``$\boldsymbol{f}$ is separated by at least one bitflip
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from all illegal configurations''
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\end{itemize}
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\end{minipage}%
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\hfill%
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\begin{minipage}[c]{0.4\linewidth}
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\begin{figure}[H]
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{245}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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% Cube
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\draw[dashed] (0, 0, 0) -- (2, 0, 0);
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\draw[dashed] (2, 0, 0) -- (2, 0, 2);
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\draw[] (2, 0, 2) -- (0, 0, 2);
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\draw[] (0, 0, 2) -- (0, 0, 0);
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\draw[] (0, 2, 0) -- (2, 2, 0);
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\draw[] (2, 2, 0) -- (2, 2, 2);
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\draw[] (2, 2, 2) -- (0, 2, 2);
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\draw[] (0, 2, 2) -- (0, 2, 0);
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\draw[] (0, 0, 0) -- (0, 2, 0);
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\draw[dashed] (2, 0, 0) -- (2, 2, 0);
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\draw[] (2, 0, 2) -- (2, 2, 2);
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\draw[] (0, 0, 2) -- (0, 2, 2);
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% Codeword Polytope
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 0, 2);
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (2, 2, 0);
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\draw[line width=1pt, color=KITblue] (0, 0, 0) -- (0, 2, 2);
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\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (2, 2, 0);
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\draw[line width=1pt, color=KITblue] (2, 0, 2) -- (0, 2, 2);
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\draw[line width=1pt, color=KITblue] (0, 2, 2) -- (2, 2, 0);
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% Polytope Annotations
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\node[codeword, color=KITred] (c111) at (2, 2, 2) {};% {$\left( 0, 0, 0 \right) $};
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\node[codeword, color=KITred] (c001) at (0, 0, 2) {};% {$\left( 1, 0, 1 \right) $};
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\node[codeword, color=KITred] (c100) at (2, 0, 0) {};% {$\left( 1, 1, 0 \right) $};
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\node[codeword, color=KITred] (c010) at (0, 2, 0) {};% {$\left( 0, 1, 1 \right) $};
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\node[color=KITred, left=0cm of c111] {$\left( 1, 1, 1 \right) $};
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\node[color=KITred, right=0cm of c001] {$\left( 0, 0, 1 \right) $};
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\node[color=KITred, right=0.35cm of c100] {$\left( 1, 0, 0 \right) $};
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\node[color=KITred, below=0cm of c010] {$\left( 0, 1, 0 \right) $};
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\end{tikzpicture}
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\caption{Relaxed polytope for $n=3$}
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\end{figure}
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\end{minipage}
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\todo{How is this a relaxation and not just an alternative formulation?
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We have just switched out valid codewords for invalid ones}
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\todo{Is LP Relaxation relevant as theoretical background?}
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\end{frame}
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