Add syndrome update equation

src/thesis/chapters/4_decoding_under_dems.tex

@@ -376,8 +376,8 @@ explicitly work with the \ac{dem} formalism.
 % }
 
 %%%%%%%%%%%%%%%%
-\subsection{Algorithm}
-\label{subsec:Algorithm}
+\subsection{Window Splitting and Sequential Sliding-Window Decoding}
+\label{subsec:Window Splitting and Sequential Sliding-Window Decoding}
 
 In this section, we will examine the methodology by which a detector
 error matrix is divided into overlapping windows.
@@ -634,32 +634,31 @@ and after decoding all windows we will therefore have committed all \acp{vn}.
     \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}
-\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{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
-%     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}
+\Cref{fig:vis_rep} illustrates the meaning of the various sets of nodes.
+We can also see a particular point we have to be careful about when
+moving on to decode the next window.
+While the \acp{vn} in $\mathcal{J}_\text{commit}^{(\ell)}$ have no
+bearing on the further decoding process, the values commit for the
+\acp{vn} in $\mathcal{I}_\text{commit}^{(\ell)}$ do.
+This is the case because these \acp{vn} have neighboring \acp{cn} in
+the next window.
+The part of the detector error matrix $\bm{H}_\text{DEM}$ describing
+these connections is
+$\bm{H}_\text{overlap}^{(\ell)} =
+\left(\bm{H}_\text{DEM}\right)_{\mathcal{J}_\text{overlap}^{(\ell)},
+\mathcal{I}_\text{commit}^{(\ell)}}$.
+We have to account for this fact by updating the syndrome $\bm{s}$
+based on the committed bit values.
+Specifically, if $\bm{e}_\text{commit}^{(\ell)}$ describes the error
+estimates commited after decoding window $\ell$, we have to set
+\begin{align*}
+    \bm{s}_{\mathcal{J}_\text{overlap}^{(\ell)}} =
+    \bm{H}_\text{overlap}^{(\ell)}
+    \left( \bm{e}_\text{commit}^{(\ell)} \right)^\text{T}
+    .%
+\end{align*}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Warm-Start Sliding-Window Decoding}