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Replaced Q with overline{Q}

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Andreas Tsouchlos 2023-02-18 17:35:57 +01:00
parent 40e731d111
commit a14ad6d9e5
1 changed files with 7 additions and 7 deletions
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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.
By observing that each check-node defines its own local single parity-check
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.
This consideration leads to the following constraints:%
%
@ -245,7 +245,7 @@ This consideration leads to the following constraints:%
\ldots
.\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.
Figure \ref{fig:dec:poly:exact} shows the codeword polytope
$\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.
Figures \ref{fig:dec:poly:local1} and \ref{fig:dec:poly:local2} show the local
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}.
%
@ -537,7 +537,7 @@ Their intersection, the relaxed codeword polytope $Q$, is shown in figure
{$\left( 1, \frac{1}{2}, \frac{1}{2} \right) $};
\end{tikzpicture}
\caption{Relaxed codeword polytope $Q$}
\caption{Relaxed codeword polytope $\overline{Q}$}
\label{fig:dec:poly:relaxed}
\end{subfigure}
\end{subfigure}
@ -548,12 +548,12 @@ Their intersection, the relaxed codeword polytope $Q$, is shown in figure
\end{figure}
\noindent%
It can be seen, that the relaxed codeword polytope $Q$ introduces fractional
vertices;
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 $Q$ 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.
The resulting formulation of the relaxed optimization problem