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Moved polytope example figure

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Andreas Tsouchlos 2023-02-20 08:21:45 +01:00
parent e3cd531767
commit 8b3c322ade
1 changed files with 15 additions and 15 deletions
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latex/thesis/chapters/decoding_techniques.tex
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@ -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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