From 8b3c322adeda72dd92f46ca44ff2ecfa3f4d5e83 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Mon, 20 Feb 2023 08:21:45 +0100 Subject: [PATCH] Moved polytope example figure --- latex/thesis/chapters/decoding_techniques.tex | 30 +++++++++---------- 1 file changed, 15 insertions(+), 15 deletions(-) diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex index a0b320f..6d27d76 100644 --- a/latex/thesis/chapters/decoding_techniques.tex +++ b/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*}% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%