Rewrite VN and CN set definition text; Fix earlier TODOs

src/thesis/chapters/4_decoding_under_dems.tex

@@ -146,7 +146,7 @@ Each of these windows is then decoded separately.
 
         \node[literature, below right=1mm and -12mm of qldpc] (huang) {\cite{huang_improved_2023},\cite{huang_increasing_2024}};
         \node[literature, below=of huang]                     (gong)  {\cite{gong_toward_2024}};
-        \node[literature, below=of gong]                     (kang)   {\cite{kang_quits_2025}};
+        \node[literature, below=of gong]                      (kang)  {\cite{kang_quits_2025}};
 
         \coordinate (code-anchor)  at ($(code.south) + (-2mm,0)$);
         \coordinate (top-anchor)   at ($(top.south) + (-5mm,0)$);
@@ -397,19 +397,25 @@ This block-diagonal structure introduces some locality in the
 interdependence between \acp{vn} and \acp{cn}.
 For each local set of \acp{vn}, there is only a local set of connected \acp{cn}.
 We exploit this fact by cutting the matrix into overlapping windows.
-\Cref{fig:windowing_pcm} depicts this process.
+\Cref{fig:windowing_pcm} depicts this process using the $\llbracket
+72, 6, 6 \rrbracket$ BB code as an example.
 
 % High-level overview
 
 How the locality is leveraged can be understood by considering the
 decoding process.
 After decoding a window, there is a subset of \acp{cn} that no longer
-contribute to the decoding process, as they do not share any \acp{vn}
+contribute to decoding, as they are not connected to any \acp{vn}
-with the \acp{cn} of subsequent windows.\\
+considered for the subsequent windows.
-\content{Commit VNs}
+We call the set of \acp{vn} connected to those \acp{cn} the
-\content{Benefit of this approach (as stated above: earlier decoding start)}
+\emph{commit region} and we wish to commit them before moving to the
+next window, i.e., fix the values we estimate for the corresponding bits.
+As mentioned above, the benefit of this sequential sliding-window
+decoding approach
+is that the decoding process can begin as soon as the syndrome
+measurements for the first window are complete.
 
-% W and F
+% W and F and why we look at rows, not columns
 
 There are two degrees of freedom in how we perform the windowing.
 The \emph{window size} $W \in \mathbb{N}$ represents the number of
@@ -417,10 +423,7 @@ syndrome extraction rounds lumped into one window.
 The \emph{step size} $F \in \mathbb{N}$ represents the number of
 syndrome extraction rounds passed over before starting the next window.
 $W$ controls the size of the windows while $F$ controls the overlap
-between windows.
+between them.
-
-% Why we look at rows, not columns
-
 As illustrated in \Cref{fig:windowing_pcm}, $W$ and $F$ control the
 window dimensions and locactions by defining the related \acp{cn},
 not the \acp{vn}.
@@ -429,28 +432,6 @@ by the choice of the underlying code and the number of syndrome
 measurement rounds, the number of \acp{vn} depends on the noise model
 and is difficult to predict beforehand.
 
-% How we get the corresponding rows and columns
-
-\content{How we get the rows}
-\content{Explain how we get the columns once we know the rows}
-\content{\textbf{General note}: Mathematical definitions where possible}
-
-% 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}
-
-% Complete process
-
-\content{(?) Proper algorithm definition?}
-\content{1. Decode window}
-\content{2. Commit VN estimates}
-\content{3. Update syndrome}
-\content{4. Decode next window}
-\content{(?) Explicitly mention we don't reuse existing messages?}
-
 \begin{figure}[t]
     \centering
 
@@ -481,12 +462,127 @@ and is difficult to predict beforehand.
     \vspace*{10mm}
 
     \caption{
-        Visualization of the windowing process on the detector
+        Visualization of the windowing process on a detector
-        error matrix.
+        error matrix generated from the $\llbracket 72, 6, 6
+        \rrbracket$ BB code.
     }
     \label{fig:windowing_pcm}
 \end{figure}
 
+% Notation recap
+
+We briefly reintroduce the notation important for the definition of the windows.
+We use the variables $n,m \in \mathbb{N}$ to describe the number of
+\acp{vn} and \acp{cn} respectively.
+We index the \acp{vn} using the variable $i \in \mathcal{I} := \left\{
+1,\ldots,n \right\}$ and the \acp{cn} using the variable $j \in
+\mathcal{J} := \left\{ 1, \ldots, m \right\}$.
+Finally, we call $\mathcal{N}_\text{V}(i) = \left\{ i\in \mathcal{I}:
+\bm{H}_{j,i} = 1 \right\}$ and $\mathcal{N}_\text{C}(j) := \left\{ j
+\in \mathcal{J} : \bm{H}_{j,i} = 1 \right\}$ the neighborhoods of the
+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.
+
+% How we get the corresponding rows
+
+We begin by describing the sets of \acp{cn} relevant to each window.
+For indexing, we use the variable $\ell \in \left\{
+0,\ldots,n_\text{win} - 1 \right\}$, where $n_\text{win} \in \mathbb{N}$
+is the number of windows.
+Because we defined the step size $F$ as the number of syndrome
+extraction rounds to skip, the first \ac{cn} of window $\ell$ should have index
+$\ell F m$.
+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*}
+}%
+$\mathcal{J}_\text{win}^{(\ell)}$ 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.
+
+% How we get the corresponding columns
+
+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
+\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*}
+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
+$i_\text{max}(\mathcal{J}_\text{commit}^{\ell})$.
+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*}
+}%
+Note that we have
+\begin{align*}
+    \bigcup_{\ell=0}^{n_\text{win}-1}
+    \mathcal{I}_\text{commit}^{(\ell)} = \mathcal{I}
+\end{align*}
+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}
+
+% Complete process
+
+\content{(?) Proper algorithm definition?}
+\content{1. Decode window}
+\content{2. Commit VN estimates}
+\content{3. Update syndrome}
+\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?
 % \content{Possibly go into the fact that current sliding-window
 %     approaches don't differentiate clearly between the sliding-window