Added descriptive text in appendix and moved around the order of figures
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\chapter{A Comparison of the Behaviour of Proximal Decoding for Various Codes}
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In this chapter, the results obtained with the proximal decoding algorithm,
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shown in chapter \ref{chapter:proximal_decoding},
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are revisited and expanded to a number of different codes.
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These codes include a BCH code with $n=31$, four \ac{LDPC} codes with
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lengths $n=96, n=204, n=204$ and $n=408$, respectively, and a progressive
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edge growth construction \ac{LDPC} code with $n=504$.
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All \ac{LDPC} codes have been taken from \cite{mackay_enc}.
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This selection of codes was chosen as it comprises a variety of codes of
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different lengths, rates and construction schemes.
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Figure \ref{fig:prox:results_3d_multiple} depicts how the choice of the
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parameter $\gamma$ affects the decoding performance.
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The \ac{BER} curves for three values of $\gamma$ are highlighted in each case:
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$\gamma = 0.01, \gamma=0.05$ and $\gamma=0.15$.
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These are the same values chosen by Wadayama et al. for the presentation of
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their results in \cite[Sec. 5.1]{proximal_paper}
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and they seem to suitably describe the behaviour of the decoding performance
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over a wide range of values for $\gamma$.
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The parameter $\omega$ is chosen to be $0.05$ and the maximum number of
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iterations performed $K$ is 100.
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$\eta$ is set to $1.5$.
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In figure \ref{fig:prox:gamma_omega_multiple}, the relationship between
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$\gamma$ and $\omega$ and how they affect the \ac{BER} is shown.
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This time a constant $E_b / N_0$ of $\SI{4}{dB}$ is chosen.
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Again, the maximum number of iterations is set to $K=100$ and $\eta$
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is set to $1.5$.
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The possible gain in the decoding performance achievable using the
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improved proximal decoding algorithm presented in section
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\ref{sec:prox:Improved Implementation} is illustrated in figure
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\ref{fig:prox:improved:comp}.
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The parameters chosen are the same as for the reconstruction of the results
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of Wadayama et al. in section \ref{sec:prox:Analysis and Simulation Results},
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i.e., $\omega = 0.05, K=100, \eta=1.5$ and
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$\gamma \in \left\{ 0.01, 0.05, 0.15 \right\}$.
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\begin{figure}[H]
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\centering
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@ -228,10 +267,238 @@
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\end{subfigure}
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\caption{BER for $\omega = 0.05, K=100$ (different codes)}
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\caption{Dependence of the \ac{BER} on the value of the parameter $\gamma$ for various codes}
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\label{fig:prox:results_3d_multiple}
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\end{figure}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_963965.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=96, k=48$
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\cite[\text{96.3.965}]{mackay_enc}}
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\end{subfigure}%
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\hfill
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_bch_31_26.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{BCH code with $n=31, k=26$\\[2\baselineskip]}
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\end{subfigure}%
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\vspace{3mm}
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_20433484.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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\end{subfigure}%
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\hfill
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_20455187.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 5, 10 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.55.187}]{mackay_enc}}
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\end{subfigure}%
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\vspace{3mm}
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_40833844.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=408, k=204$
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\cite[\text{408.33.844}]{mackay_enc}}
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\end{subfigure}%
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\hfill
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_pegreg252x504.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{LDPC code (Progressive Edge Growth Construction) with $n=504, k=252$
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\cite[\text{PEGReg252x504}]{mackay_enc}}
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\end{subfigure}%
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\vspace{5mm}
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\begin{subfigure}{\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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hide axis,
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scale only axis,
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height=0pt,
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width=0pt,
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colormap/viridis,
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colorbar horizontal,
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point meta min=-5.7,
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point meta max=-0.5,
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colorbar style={
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title={BER},
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width=10cm,
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xtick={-5,-4,...,-1},
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xticklabels={$10^{-5}$,$10^{-4}$,$10^{-3}$,$10^{-2}$,$10^{-1}$}
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}]
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\addplot [draw=none] coordinates {(0,0)};
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\end{axis}
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\end{tikzpicture}
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\end{subfigure}%
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\caption{The \ac{BER} as a function of $\gamma$ and $\omega$ for various codes}
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\label{fig:prox:gamma_omega_multiple}
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\end{figure}
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\begin{figure}[H]
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\centering
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@ -542,234 +809,6 @@
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\label{fig:prox:improved:comp}
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\end{figure}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_963965.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=96, k=48$
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\cite[\text{96.3.965}]{mackay_enc}}
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\end{subfigure}%
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\hfill
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_bch_31_26.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{BCH code with $n=31, k=26$\\[2\baselineskip]}
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\end{subfigure}%
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\vspace{3mm}
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_20433484.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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\end{subfigure}%
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\hfill
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_20455187.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 5, 10 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.55.187}]{mackay_enc}}
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\end{subfigure}%
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\vspace{3mm}
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\begin{subfigure}[t]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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colormap/viridis,
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xlabel={$\omega$}, ylabel={$\gamma$},
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at={(0,0)}, view={0}{90},
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zmode=log,
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ytick={0, 0.05, 0.1, 0.15},
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yticklabels={0, 0.05, 0.1, 0.15},
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xtick={0.05, 0.1, 0.15, 0.2},
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xticklabels={0.05, 0.1, 0.15, 0.2},
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width=\textwidth,
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height=0.75\textwidth,
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point meta min=-5.7,
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point meta max=-0.5,
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]
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\addplot3[
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surf,
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shader=flat,
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mesh/rows=17, mesh/cols=10,
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]
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table [col sep=comma, x=omega, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_gamma_omega_40833844.csv};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
|
||||
\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=408, k=204$
|
||||
\cite[\text{408.33.844}]{mackay_enc}}
|
||||
\end{subfigure}%
|
||||
\hfill
|
||||
\begin{subfigure}[t]{0.48\textwidth}
|
||||
\centering
|
||||
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[
|
||||
colormap/viridis,
|
||||
xlabel={$\omega$}, ylabel={$\gamma$},
|
||||
at={(0,0)}, view={0}{90},
|
||||
zmode=log,
|
||||
ytick={0, 0.05, 0.1, 0.15},
|
||||
yticklabels={0, 0.05, 0.1, 0.15},
|
||||
xtick={0.05, 0.1, 0.15, 0.2},
|
||||
xticklabels={0.05, 0.1, 0.15, 0.2},
|
||||
width=\textwidth,
|
||||
height=0.75\textwidth,
|
||||
point meta min=-5.7,
|
||||
point meta max=-0.5,
|
||||
]
|
||||
\addplot3[
|
||||
surf,
|
||||
shader=flat,
|
||||
mesh/rows=17, mesh/cols=10,
|
||||
]
|
||||
table [col sep=comma, x=omega, y=gamma, z=BER]
|
||||
{res/proximal/2d_ber_fer_dfr_gamma_omega_pegreg252x504.csv};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
|
||||
\caption{LDPC code (Progressive Edge Growth Construction) with $n=504, k=252$
|
||||
\cite[\text{PEGReg252x504}]{mackay_enc}}
|
||||
\end{subfigure}%
|
||||
|
||||
\vspace{5mm}
|
||||
|
||||
\begin{subfigure}{\textwidth}
|
||||
\centering
|
||||
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[
|
||||
hide axis,
|
||||
scale only axis,
|
||||
height=0pt,
|
||||
width=0pt,
|
||||
colormap/viridis,
|
||||
colorbar horizontal,
|
||||
point meta min=-5.7,
|
||||
point meta max=-0.5,
|
||||
colorbar style={
|
||||
title={BER},
|
||||
width=10cm,
|
||||
xtick={-5,-4,...,-1},
|
||||
xticklabels={$10^{-5}$,$10^{-4}$,$10^{-3}$,$10^{-2}$,$10^{-1}$}
|
||||
}]
|
||||
\addplot [draw=none] coordinates {(0,0)};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\end{subfigure}%
|
||||
|
||||
\caption{}
|
||||
\label{fig:prox:gamma_omega_multiple}
|
||||
\end{figure}
|
||||
|
||||
|
||||
%\chapter{\acs{LP} Decoding using \acs{ADMM} as a Proximal Algorithm}%
|
||||
%\label{chapter:LD Decoding using ADMM as a Proximal Algorithm}
|
||||
|
||||
@ -449,8 +449,8 @@ in each case.
|
||||
xlabel={$E_b / N_0$ (dB)},
|
||||
ylabel={$\gamma$},
|
||||
zlabel={BER},
|
||||
%legend pos=outer north east,
|
||||
legend style={at={(0.5,-0.7)},anchor=south},
|
||||
legend pos=outer north east,
|
||||
%legend style={at={(0.5,-0.7)},anchor=south},
|
||||
ytick={0, 0.05, 0.1, 0.15},
|
||||
width=0.6\textwidth,
|
||||
height=0.45\textwidth,]
|
||||
@ -528,7 +528,7 @@ depicted in figure \ref{fig:prox:gamma_omega_multiple}.
|
||||
point meta min=-5.7,
|
||||
point meta max=-0.5,
|
||||
colorbar style={
|
||||
title={BER},
|
||||
title={$E_b / N_0$},
|
||||
ytick={-5,-4,...,-1},
|
||||
yticklabels={$10^{-5}$,$10^{-4}$,$10^{-3}$,$10^{-2}$,$10^{-1}$}
|
||||
}]
|
||||
@ -548,7 +548,7 @@ depicted in figure \ref{fig:prox:gamma_omega_multiple}.
|
||||
\end{figure}%
|
||||
%
|
||||
\footnotetext{(3,6) regular \ac{LDPC} code with n = 204, k = 102
|
||||
\cite[\text{204.33.484}]{mackay_enc}; $SNR=\SI{4}{dB}, K=100, \eta=1.5$
|
||||
\cite[\text{204.33.484}]{mackay_enc}; $E_b / N_0=\SI{4}{dB}, K=100, \eta=1.5$
|
||||
}%
|
||||
%
|
||||
|
||||
@ -561,7 +561,7 @@ different \acp{SNR}.
|
||||
The plots have been generated by averaging the error over TODO decodings.
|
||||
As some decodings go one for more iterations than others, the number of values
|
||||
which are averaged for each datapoints vary.
|
||||
This explains the bump observable around $k=\text{TODO}$, since after
|
||||
This explains the bump visible around $k=\text{TODO}$, since after
|
||||
this point more and more correct decodings converge and stop iterating,
|
||||
leaving more and more faulty ones to be averaged.
|
||||
Remarkably, the \ac{SNR} seems to not have any impact on the number of
|
||||
|
||||
Loading…
Reference in New Issue
Block a user