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Finish index definitions

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Andreas Tsouchlos
2026-05-01 18:08:57 +02:00
parent 3b7618e1d1
commit 4555570665
1 changed files with 39 additions and 39 deletions
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78
src/thesis/chapters/4_decoding_under_dems.tex
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@@ -482,6 +482,8 @@ Finally, we call $\mathcal{N}_\text{V}(i) = \left\{ i\in \mathcal{I}:
corresponding nodes.
In this case, we take $\bm{H} \in \mathbb{F}_2^{m\times n}$ to be the
check matrix of the underlying code, from which the \ac{dem} was generated.
We use $m_\text{DEM}, \mathcal{I}_\text{DEM}$, and $\mathcal{J}_\text{DEM}$
to refer to the respective values defined from the detector error matrix.
% How we get the corresponding rows
@@ -495,23 +497,17 @@ Similarly, because of the way we defined the step size $W$, the
number of \acp{cn} should be $Wm$ for all but the last window.
The number of \acp{cn} in the last window may differ if there are
not enough \acp{cn} left to completely fill it.
We thus define%
\footnote{
Note that the inequality is written in terms of $j-1$, not just $j$.
This is done to transform between zero-based and one-based indexing.
}
\red{
\begin{align*}
\mathcal{J}_\text{win}^{(\ell)}
&:= \left\{ j\in \mathcal{J}:~ \ell F m \le j - 1 < (\ell F +
W)\cdot m \right\} \\[2mm]
& \hspace{30mm} \text{and} \\[2mm]
\mathcal{J}_\text{commit}^{(\ell)}
&:= \left\{ j\in \mathcal{J}:~ \ell F m \le j - 1 < (\ell + 1)\cdot
F m \right\}
.%
\end{align*}
}%
We thus define
\begin{align*}
\mathcal{J}_\text{win}^{(\ell)} &:= \left\{ j\in \mathcal{J}_\text{DEM}:~
\ell F m \le j < \min \left\{m_\text{DEM}, (\ell F + W) m \right\}
\right\} \\[2mm]
& \hspace{30mm} \text{and} \\[2mm]
\mathcal{J}_\text{commit}^{(\ell)} &:= \left\{ j\in \mathcal{J}_\text{DEM}:~
\ell F m \le j < \min \left\{m_\text{DEM}, (\ell + 1) F m \right\}
\right\}
.%
\end{align*}
$\mathcal{J}_\text{win}^{(\ell)}$ 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
@@ -522,39 +518,44 @@ that do not contribute to the next window and whose neighboring
We can now turn our attention to defining the sets of \acp{vn} relevant
to each window.
We first introduce a helper function $i_\text{max} :
\mathcal{P}(\mathbb{N}) \mapsto \mathbb{N}$, which takes a set of
\mathcal{P}(\mathbb{N}) \to \mathbb{N}$, which takes a set of
\ac{cn} indices and returns the largest neighboring \ac{vn} index.
We define
\begin{align*}
i_\text{max}\left( \mathcal{S} \right) := \max \left\{ i\in
\mathcal{N}_\text{C}(j) : j\in \mathcal{S} \right\}
.
,
\end{align*}
where we set $i_\text{max} (\emptyset) = -1$ by convention%
\footnote{
This has the effect of later automatically setting the lower
bounds for the indices in $\mathcal{I}_\text{commit}^{(\ell)}$
and $\mathcal{I}_\text{win}^{(\ell)}$ appropriately.
}%
.
The commit region of window $\ell$ should include all of the \acp{vn}
neighboring any of the \acp{cn} in $\mathcal{J}_\text{commit}^{(\ell)}$.
Consequently, the maximum index of the \acp{vn} we consider should be
$i_\text{max}(\mathcal{J}_\text{commit}^{\ell})$.
$i_\text{max}(\mathcal{J}_\text{commit}^{(\ell)})$.
Additionally, the set of \acp{vn} comitted in the next window should
start immediately afterwards.
We thus define
\red{
\begin{align*}
\mathcal{I}_\text{commit}^{(\ell)}
&:= \left\{i \in \mathcal{I} :~
i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
< i - 1 \le
i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell)} \right)
\right\}\\[2mm]
& \hspace{39mm} \text{and} \\[2mm]
\mathcal{I}_\text{win}^{(\ell)}
&:= \left\{i \in \mathcal{I} :~
i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
< i - 1 \le
i_\text{max}\left( \mathcal{J}_\text{win}^{(\ell)} \right)
\right\}
.%
\end{align*}
}%
\begin{align*}
\mathcal{I}_\text{commit}^{(\ell)}
&:= \left\{i \in \mathcal{I}_\text{DEM} :~
i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
< i \le
i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell)} \right)
\right\}\\[2mm]
& \hspace{39mm} \text{and} \\[2mm]
\mathcal{I}_\text{win}^{(\ell)}
&:= \left\{i \in \mathcal{I}_\text{DEM} :~
i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
< i \le
i_\text{max}\left( \mathcal{J}_\text{win}^{(\ell)} \right)
\right\}
.%
\end{align*}
Note that we have
\begin{align*}
\bigcup_{\ell=0}^{n_\text{win}-1}
@@ -564,7 +565,6 @@ and after decoding all windows we will therefore have committed all \acp{vn}.
% Syndrome update
\content{Explain commit region}
\content{Why we need to update the syndrome}
\content{How we update the syndrome}
\content{\textbf{General note}: Mathematical definitions where possible}