Wwrite a few paragraphs on the window generation/decoding

src/thesis/chapters/4_decoding_under_dems.tex

@@ -376,33 +376,62 @@ explicitly work with the \ac{dem} formalism.
 % }
 
 %%%%%%%%%%%%%%%%
-\subsection{Window Generation}
-\label{subsec:Window Generation}
+\subsection{Algorithm}
+\label{subsec:Algorithm}
 
 In this section, we will examine the methodology by which a detector
 error matrix is divided into overlapping windows.
 The algorithm detailed here follows \cite{kang_quits_2025}, whose
 work is in turn based on \cite{huang_increasing_2024}.
 
-\content{Possibly go into the fact that current sliding-window
-    approaches don't differentiate clearly between the sliding-window
-    part and the decoder part. This work aims to extend the
-    sliding-window part in a general fashion that is compatible with many
-    different decoder parts. Combine this with QUITS modular structure
-for sliding window decoding}
+% Very high-level overview
 
-% High-level overview of Sliding-Window decoding
+Sliding-window decoding is made possible by the time-like structure
+of the syndrome extraction circuitry.
+This is epecially clearly visible under the \ac{dem} formalism, where
+this manifests as a block-diagonal structure of the detector
+error matrix $\bm{H}$.
+Note that this presupposes a choice of detectors as seen in
+\Cref{subsec:Detector Error Matrix}.
+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.
 
-\content{Benefits of sliding-window decoding (lower latency due to
-earlier decoding start)}
-\content{Why it works (block diagonal structure $\rightarrow$ ``Done
-with processing'' some VNs)}
+% High-level overview
 
-% Detailed explanation of sliding-window decoding
+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}
+with the \acp{cn} of subsequent windows.\\
+\content{Commit VNs}
+\content{Benefit of this approach (as stated above: earlier decoding start)}
 
-\content{We look at rows not columns}
-\content{Define W}
-\content{Define F}
+% W and F
+
+There are two degrees of freedom in how we perform the windowing.
+The \emph{window size} $W \in \mathbb{N}$ represents the number of
+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.
+
+% 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}.
+This is because while the number of overall \acp{cn} is only affected
+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}
 
@@ -415,6 +444,7 @@ with processing'' some VNs)}
 
 % Complete process
 
+\content{(?) Proper algorithm definition?}
 \content{1. Decode window}
 \content{2. Commit VN estimates}
 \content{3. Update syndrome}
@@ -457,6 +487,14 @@ with processing'' some VNs)}
     \label{fig:windowing_pcm}
 \end{figure}
 
+% TODO: Do I need this?
+% \content{Possibly go into the fact that current sliding-window
+%     approaches don't differentiate clearly between the sliding-window
+%     part and the decoder part. This work aims to extend the
+%     sliding-window part in a general fashion that is compatible with many
+%     different decoder parts. Combine this with QUITS modular structure
+% for sliding window decoding}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Warm-Start Sliding-Window Decoding}
 \label{sec:warm_start_bp}