Moved polytope example figure

latex/thesis/chapters/decoding_techniques.tex

@@ -276,21 +276,6 @@ Figure \ref{fig:dec:poly:local} shows the local codeword polytope of each check
 node.
 Their intersection, the relaxed codeword polytope $\overline{Q}$, is shown in
 figure \ref{fig:dec:poly:relaxed}.
-It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces
-vertices with fractional values;
-these represent erroneous non-codeword solutions to the linear program and
-correspond to the so-called \textit{pseudocodewords} introduced in
-\cite{feldman_paper}.
-However, since for \ac{LDPC} codes $\overline{Q}$ scales linearly with $n$ instead of
-exponentially, it is a lot more tractable for practical applications.
-
-The resulting formulation of the relaxed optimization problem is the following:%
-%
-\begin{align*}
-    \text{minimize }\hspace{2mm} &\sum_{i=1}^{n} \gamma_i c_i \\
-    \text{subject to }\hspace{2mm} &\boldsymbol{T}_j \boldsymbol{c} \in \mathcal{P}_{d_j},
-        \hspace{5mm}j\in\mathcal{J}
-.\end{align*}%
 %
 %
 %
@@ -589,6 +574,21 @@ The resulting formulation of the relaxed optimization problem is the following:%
     \label{fig:dec:poly}
 \end{figure}%
 %
+It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces
+vertices with fractional values;
+these represent erroneous non-codeword solutions to the linear program and
+correspond to the so-called \textit{pseudocodewords} introduced in
+\cite{feldman_paper}.
+However, since for \ac{LDPC} codes $\overline{Q}$ scales linearly with $n$ instead of
+exponentially, it is a lot more tractable for practical applications.
+
+The resulting formulation of the relaxed optimization problem is the following:%
+%
+\begin{align*}
+    \text{minimize }\hspace{2mm} &\sum_{i=1}^{n} \gamma_i c_i \\
+    \text{subject to }\hspace{2mm} &\boldsymbol{T}_j \boldsymbol{c} \in \mathcal{P}_{d_j},
+        \hspace{5mm}j\in\mathcal{J}
+.\end{align*}%
 
 
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