diff --git a/src/thesis/chapters/4_decoding_under_dems.tex b/src/thesis/chapters/4_decoding_under_dems.tex
index d212fb3..7a7fadd 100644
--- a/src/thesis/chapters/4_decoding_under_dems.tex
+++ b/src/thesis/chapters/4_decoding_under_dems.tex
@@ -628,12 +628,13 @@ and after decoding all windows we will therefore have committed all \acp{vn}.
         \node[align=center] at ($(a00)!0.5!(b01)$)
         {%
             $\bm{H}_\text{overlap}^{(\ell)}$ \\[3mm]
-            $= \left(\bm{H}_\text{DEM}\right)_{\mathcal{J}_\text{overlap}^{(\ell)},
+            $=
+            \left(\bm{H}_\text{DEM}\right)_{\mathcal{J}_\text{overlap}^{(\ell)},
             \mathcal{I}_\text{commit}^{(\ell)}}$%
         };
     \end{tikzpicture}
 
-    \caption{Visual representation of notation for window splitting.}
+    \caption{Visual representation of notation used for window splitting.}
     \label{fig:vis_rep}
 \end{figure}
 
@@ -669,25 +670,27 @@ estimates commited after decoding window $\ell$, we have to set
 
 % Intro
 
-\content{Change view from PCM to Tanner graph}
-\content{Call attention to SC-LDPC-like structure}
-\content{High-level overview of modification}
+The sliding-window structure visible in \Cref{fig:windowing_pcm} is
+highly reminicent of the way \ac{sc}-\ac{ldpc} codes are decoded.
+Switching our viewpoint to the Tanner graph depicted in
+\Cref{fig:windowing_tanner}, however, we can see an important
+difference between \ac{sc}-\ac{ldpc} decoding and the
+sliding-window decoding procedure detailed above.
+While the windowing process is similar, the algorithm above
+reinitializes the decoder to start from a clean state when moving to
+the next window.
+It therefore does not make use of the integral property of
+\ac{sc}-\ac{ldpc} decoding of exploiting the spatially coupled
+structure by passing soft information from earlier to later spatial positions.
 
-% Warm-Start decoding for BP
+We propose a modification to the procedure detailed in
+\Cref{subsec:Window Splitting and Sequential Sliding-Window Decoding}:
+Instead of zero-initializing the \ac{bp} messages of the next window,
+we perform a \emph{warm start} by initializing the messages in the
+overlapping region to the values last held during the decoding of the
+previous window.
 
-\content{Pass messages to next window}
-\content{(?) Explicitly mention initialization using only CN->VN
-messages + swapping of CN and VN update?}
-\content{(?) Algorithm}
-
-% Warm-Start decoding for BPGD
-
-\content{Modified structure of BPGD $\rightarrow$ In addition to
-messages, pass decimation info}
-\content{(?) Explicitly mention decimation info = channel llrs?}
-\content{(?) Algorithm}
-
-\begin{figure}[H]
+\begin{figure}[t]
     \centering
 
     \tikzset{
@@ -789,7 +792,18 @@ messages, pass decimation info}
     \label{fig:windowing_tanner}
 \end{figure}
 
-\begin{figure}[H]
+%%%%%%%%%%%%%%%%
+\subsection{Warm-Start Belief Propagation Decoding}
+\label{subsec:Warm-Start Belief Propagation Decoding}
+
+% Warm-Start decoding for BP
+
+\content{Pass messages to next window}
+\content{(?) Explicitly mention initialization using only CN->VN
+messages + swapping of CN and VN update?}
+\content{(?) Algorithm}
+
+\begin{figure}[t]
     \centering
 
     \tikzset{
@@ -911,7 +925,18 @@ messages, pass decimation info}
     \label{fig:messages_tanner}
 \end{figure}
 
-\begin{figure}[H]
+%%%%%%%%%%%%%%%%
+\subsection{Warm-Start Belief Propagation with Guided Decimation Decoding}
+\label{subsec:Warm-Start Belief Propagation with Guided Decimation Decoding}
+
+% Warm-Start decoding for BPGD
+
+\content{Modified structure of BPGD $\rightarrow$ In addition to
+messages, pass decimation info}
+\content{(?) Explicitly mention decimation info = channel llrs?}
+\content{(?) Algorithm}
+
+\begin{figure}[t]
     \centering
 
     \tikzset{
@@ -1045,16 +1070,122 @@ messages, pass decimation info}
     \label{fig:messages_decimation_tanner}
 \end{figure}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Numerical results}
-\label{sec:warm_start_bpgd}
+\section{Numerical Results}
+\label{sec:Numerical Results}
 
 % Intro
 
-\content{Some info on used code (what it is, why it was chosen)}
-\content{Some info on simulation setup (Stim, circuit-level noise,
-standard circuit-based depolarizing noise model, etc.)}
-\content{All datapoints generated with at least 100 error frames}
+In this section, we perform numerical experiments to evaluate the
+modification to sliding-window decoding we introduced in
+\Cref{sec:warm_start_bp}.
+We chose to carry out our simulations on \ac{bb} codes, as they have
+recently emerged as particularly promising candidates for practical
+\ac{qec}, offering high encoding rates and large minimum distances
+while admitting short-depth syndrome extraction circuits
+\cite[Sec.~1]{bravyi_high-threshold_2024}.
+Specifically, we chose the $\llbracket 144, 12, 12 \rrbracket$ BB
+code, as it represents a good trade-off between code size and
+simulation tractability \cite{gong_toward_2024}.
+We employ standard circuit-based depolarizing noise as described in
+\Cref{subsec:Choice of Noise Model}, and report performance in terms
+of the per-round \ac{ler} as defined in
+\Cref{subsec:Per-Round Logical Error Rate}.
+All datapoints have been generated by simulating at least $200$
+logical error events.
+
+For the practical aspects of implementation, several layers of
+abstraction must be considered.
+The lowest layer is the circuit-level simulator.
+This serves as the backbone of all further simulations, handling the
+quantum mechanical aspects of the system, including the modeling of
+noise on gates, idling qubits, and measurements according to the
+chosen noise model.
+
+Moving one level of abstraction higher, the syndrome extraction
+circuit itself must be generated.
+This entails defining the gate schedule for the ancilla measurements
+and specifying the error locations introduced by the chosen noise
+model, both of which depend on the code and noise model in question.
+
+Even further up, given an already constructed syndrome extraction
+circuit and the resulting \acf{dem}, we must split the detector error
+matrix into separate windows and manage the interplay between the
+inner decoders acting on those individual windows.
+
+Finally, we require the decoder itself, which operates on a
+\acf{pcm} and a syndrome, with no dependence on the complexity of the
+layers below.
+
+In our implementation, Stim \cite{gidney_stim_2021} served as the
+circuit-level simulator, chosen for its efficiency and native support
+for the \ac{dem} formalism.
+For the circuit generation, we employed utilities from QUITS
+\cite{kang_quits_2025}, which provides syndrome extraction circuitry
+generation for a number of different \ac{qldpc} codes.
+We initially created a Python implementation, which used QUITS for the window
+splitting and subsequent sliding-window decoding as well.
+The \ac{bp} and \ac{bpgd} decoders were also initially implemented in Python.
+After a preliminary investigation, we opted for a complete
+reimplementation in Rust to achieve higher simulation speeds due to
+the compiled nature of the language.
+We reimplemented both the window splitting and the decoders themselves.
+
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Numerical results}
+%
+% % Intro
+%
+% In this section, we perform numerical experiments to evaluate the
+% modification to sliding-window decoding we introduced in
+% \Cref{sec:warm_start_bp}.
+% We chose to carry out our simulations on \ac{bb} codes, as they
+% \red{[something about qldpc codes being a hot topic in the literature
+% currently because of some promising properties]}.
+% Specifically, we chose the $\llbracket 144, 12, 12 \rrbracket$ BB
+% code, as this \red{[something something]}.\\
+% \red{[Circuit-level noise]} \\
+% \red{[Per-round LER]} \\
+% All datapoints have been generated by simulating at least $200$
+% logical error events.
+%
+% For the practical aspects of implementation, several layers of
+% abstraction must be considered.
+% The lowest layer is the circuit-level simulator.
+% This serves as the backbone of all further simulations.
+% It takes care of the quantum mechanical aspects of the system.
+% \red{It is, for example, responsible for the introduction of noise
+% [rephrase this]}.
+%
+% Moving one level of abstraction higher, aside from the circuit
+% simulation itself, the circuit also has to be generated.
+% E.g., the syndrome extraction circuitry must be defined and possible
+% sources of noise must be modeled.
+% This heavily depends on the code in question and the chosen noise model.
+% \red{[Find something more to say]}
+%
+% Even further up, we have already defined syndrome extraction
+% circuitry and built the \acf{dem}.
+% We must now split the detector error matrix into separate windows and
+% manage the interplay of the inner decoders acting on the individual
+% windows themselves.
+% \red{[Find something more to say]}
+%
+% Finally, we require the decoder itself.
+% This simply gets a \acf{pcm} and a syndrome with no regard
+% \red{[Rephrase this] for the complexity in the rest of the system}.
+%
+% In our implementations, Stim \cite{gidney_stim_2021} served as the
+% circuit-level simulator \red{[Possibly mention why stim was chosen]}.
+% For the circuit generation, we employed utilities from QUITS
+% \cite{kang_quits_2025}, where syndrome extraction circuitry
+% generation is implemented for a number of different \ac{qldpc} codes.
+% An initial Python implementation used QUITS for the window
+% splitting and subsequent sliding-window decoding as well.
+% The \ac{bp} and \ac{bpgd} decoders were also initially implemented in Python.
+% After a preliminary investigation, we opted for a complete
+% reimplementation in Rust to achieve higher simulation speeds due to
+% the compiled nature of the language.
+% We reimplemented both the window splitting and the decoders themselves.
 
 %%%%%%%%%%%%%%%%
 \subsection{Belief Propagation}