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Replaced Q with overline{Q}

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Andreas Tsouchlos 2023-02-18 17:35:57 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -237,7 +237,7 @@ However, since the number of constraints needed to characterize the codeword
polytope is exponential in the code length, this formulation is relaxed futher. polytope is exponential in the code length, this formulation is relaxed futher.
By observing that each check-node defines its own local single parity-check By observing that each check-node defines its own local single parity-check
code, and thus its own \textit{local codeword polytope}, code, and thus its own \textit{local codeword polytope},
the \textit{relaxed codeword polytope} $Q$ is defined as the intersection of all the \textit{relaxed codeword polytope} $\overline{Q}$ is defined as the intersection of all
local codeword polytopes. local codeword polytopes.
This consideration leads to the following constraints:% This consideration leads to the following constraints:%
% %
@ -245,7 +245,7 @@ This consideration leads to the following constraints:%
\ldots \ldots
.\end{align*} .\end{align*}
In figure \ref{fig:dec:poly} the two relaxations are compared based on an In figure \ref{fig:dec:poly}, the two relaxations are compared based on an
example code. example code.
Figure \ref{fig:dec:poly:exact} shows the codeword polytope Figure \ref{fig:dec:poly:exact} shows the codeword polytope
$\text{poly}\left( \mathcal{C} \right) $, i.e. the constraints for the $\text{poly}\left( \mathcal{C} \right) $, i.e. the constraints for the
@ -253,7 +253,7 @@ equivalent linear program to exact \ac{ML} decoding - only valid codewords are
feasible solutions. feasible solutions.
Figures \ref{fig:dec:poly:local1} and \ref{fig:dec:poly:local2} show the local Figures \ref{fig:dec:poly:local1} and \ref{fig:dec:poly:local2} show the local
codeword polytopes of each check node. codeword polytopes of each check node.
Their intersection, the relaxed codeword polytope $Q$, is shown in figure Their intersection, the relaxed codeword polytope $\overline{Q}$, is shown in figure
\ref{fig:dec:poly:relaxed}. \ref{fig:dec:poly:relaxed}.
% %
@ -537,7 +537,7 @@ Their intersection, the relaxed codeword polytope $Q$, is shown in figure
{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $}; {$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
\end{tikzpicture} \end{tikzpicture}
\caption{Relaxed codeword polytope $Q$} \caption{Relaxed codeword polytope $\overline{Q}$}
\label{fig:dec:poly:relaxed} \label{fig:dec:poly:relaxed}
\end{subfigure} \end{subfigure}
\end{subfigure} \end{subfigure}
@ -548,12 +548,12 @@ Their intersection, the relaxed codeword polytope $Q$, is shown in figure
\end{figure} \end{figure}
\noindent% \noindent%
It can be seen, that the relaxed codeword polytope $Q$ introduces fractional It can be seen, that the relaxed codeword polytope $\overline{Q}$ introduces
vertices; vertices with fractional values;
these represent erroneous non-codeword solutions to the linear program and these represent erroneous non-codeword solutions to the linear program and
correspond to the so-called \textit{pseudocodewords} introduced in correspond to the so-called \textit{pseudocodewords} introduced in
\cite{feldman_paper}. \cite{feldman_paper}.
However, since for \ac{LDPC} codes $Q$ scales linearly with $n$ instead of However, since for \ac{LDPC} codes $\overline{Q}$ scales linearly with $n$ instead of
exponentially, it is a lot more tractable for practical applications. exponentially, it is a lot more tractable for practical applications.
The resulting formulation of the relaxed optimization problem The resulting formulation of the relaxed optimization problem