an.tsouchlos/ma-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 12 Wiki Activity

Add node set Visualization figure

Browse Source
This commit is contained in:
Andreas Tsouchlos
2026-05-01 19:13:22 +02:00
parent 4555570665
commit 3d3556689e
1 changed files with 73 additions and 1 deletions
Download Patch File Download Diff File Expand all files Collapse all files

74
src/thesis/chapters/4_decoding_under_dems.tex
View File

@@ -512,6 +512,9 @@ $\mathcal{J}_\text{win}^{(\ell)}$ is the set of all \acp{cn} in the
window while $\mathcal{J}_\text{commit}^{(\ell)}$ is the set of \acp{cn}
that do not contribute to the next window and whose neighboring
\acp{vn} will thus be comitted.
We can additionally define the set of \acp{vn} that are shared between windows
$\ell$ and $\ell + 1$ as $\mathcal{J}_\text{overlap}^{(\ell)} :=
\mathcal{J}_\text{win}^{(\ell)}\setminus \mathcal{J}_\text{commit}^{(\ell)}$.
% How we get the corresponding columns
@@ -563,6 +566,76 @@ Note that we have
\end{align*}
and after decoding all windows we will therefore have committed all \acp{vn}.
\begin{figure}[t]
\centering
\begin{tikzpicture}
\def\sx{1.5}
\def\sy{1.5}
\coordinate (a00) at (0,0);
\coordinate (a01) at (0, 3*\sy);
\coordinate (a11) at (6*\sx, 3*\sy);
\coordinate (a10) at (6*\sx, 0*\sy);
\coordinate (b00) at (3.2*\sx, -1*\sy);
\coordinate (b01) at (3.2*\sx, 2*\sy);
\coordinate (b11) at (9.2*\sx, 2*\sy);
\coordinate (b10) at (9.2*\sx, -1*\sy);
\fill[gray!40] (a00) -- (a00 |- b01) -- (b01) -- (b01 |- a00) -- cycle;
\draw (a00) -- (a01) -- (a11) -- (a10) -- cycle;
\draw[densely dashed] (b00) -- (b01) -- (b11) -- (b10) -- cycle;
\draw [
decorate,
decoration={brace,amplitude=3mm,raise=1mm}
]
(a01) -- (a11)
node[midway,above,yshift=4mm]{$\mathcal{I}_\text{win}^{(\ell)}$};
\draw [
decorate,
decoration={brace,amplitude=3mm,raise=1mm}
]
(a00 -| b00) -- (a00)
node[midway,below,yshift=-4mm]{$\mathcal{I}_\text{commit}^{(\ell)}$};
\draw [
decorate,
decoration={brace,amplitude=3mm,raise=1mm}
]
(a00) -- (a01)
node[midway,xshift=-3mm,left]{$\mathcal{J}_\text{win}^{(\ell)}$};
\draw [
decorate,
decoration={brace,amplitude=3mm,raise=1mm}
]
(a11) -- (a11 |- b11)
node[midway,xshift=3mm,right]{$\mathcal{J}_\text{commit}^{(\ell)}$};
\draw [
decorate,
decoration={brace,amplitude=3mm,raise=1mm}
]
(a11 |- b11) -- (a10)
node[midway,xshift=3mm,right]{$\mathcal{J}_\text{overlap}^{(\ell)}
:= \mathcal{J}_\text{win}^{(\ell)} \setminus
\mathcal{J}_\text{commit}^{(\ell)}$};
\node at ($(a00)!0.5!(b01)$)
{$\left(\bm{H}_\text{DEM}\right)_{\mathcal{J}_\text{overlap}^{(\ell)},
\mathcal{I}_\text{commit}^{(\ell)}}$};
\end{tikzpicture}
\caption{Visual representation of notation for window splitting.}
\label{fig:vis_rep}
\end{figure}
\Cref{fig:vis_rep} illustrates the meaning of the various sets of nodes.
% Syndrome update
\content{Why we need to update the syndrome}
@@ -578,7 +651,6 @@ and after decoding all windows we will therefore have committed all \acp{vn}.
\content{4. Decode next window}
\content{(?) Explicitly mention we don't reuse existing messages?}
\content{Definition for last window is different}
\content{Double-check all definitions}
% TODO: Do I need this?