diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex
index 95bc066..b00db2e 100644
--- a/latex/thesis/chapters/decoding_techniques.tex
+++ b/latex/thesis/chapters/decoding_techniques.tex
@@ -676,29 +676,168 @@ exponentially, it is a lot more tractable for practical applications.
 
 The resulting formulation of the relaxed optimization problem becomes:%
 %
-\begin{align*}
-    \text{minimize }\hspace{2mm} &\sum_{i=1}^{n} \gamma_i \tilde{c}_i \\
-    \text{subject to }\hspace{2mm} &\boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j},
-        \hspace{5mm}j\in\mathcal{J}
-.\end{align*}%
+\begin{align}
+    \begin{aligned}
+        \text{minimize }\hspace{2mm} &\sum_{i=1}^{n} \gamma_i \tilde{c}_i \\
+        \text{subject to }\hspace{2mm} &\boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}
+        \hspace{5mm}\forall j\in\mathcal{J}.
+    \end{aligned} \label{eq:lp:relaxed_formulation}
+\end{align}%
+\todo{Rewrite sum as dot product}
+\todo{Space before $\forall$?}
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{LP Decoding using ADMM}%
 \label{sec:dec:LP Decoding using ADMM}
 
-\begin{itemize}
-    \item Why ADMM?
-        \begin{itemize}
-            \item Distributed nature, making it a competitor to BP
-                (which can also be implemented in a distributed manner)
-                \cite[Sec. I]{original_admm}
-            \item Computational performance similar to BP has been demnonstrated
-                \cite[Sec. I]{original_admm}
-        \end{itemize}
-    \item Adaptive linear programming?
-    \item How ADMM is adapted to LP decoding
-\end{itemize}
+The \ac{LP} decoding formulation in section \ref{sec:dec:Decoding using Optimization Methods}
+is a very general one that can be solved with a number of different optimization methods.
+In this work \ac{ADMM} is examined, as its distributed nature allows for a very efficient
+implementation.
+\ac{LP} decoding using \ac{ADMM} can be regarded as a message
+passing algorithm with two separate update steps that can be performed
+simulatneously;
+the resulting algorithm has a striking similarity to \ac{BP} and its computational
+complexity has been demonstrated to compare favorably to \ac{BP} \cite{original_admm},
+\cite{efficient_lp_dec_admm}.
+
+The \ac{LP} decoding problem in (\ref{eq:lp:relaxed_formulation}) can be
+slightly rewritten using the auxiliary variables
+$\boldsymbol{z}_{1:m}$:%
+%
+\begin{align}
+    \begin{aligned}
+        \begin{array}{r}
+            \text{minimize }
+        \end{array}\hspace{0.5mm}   & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}} \\
+        \begin{array}{r}
+            \text{subject to }\\
+            \phantom{te}
+        \end{array}\hspace{0.5mm}        & \setlength{\arraycolsep}{1.4pt}
+                                         \begin{array}{rl}
+                                              \boldsymbol{T}_j\tilde{\boldsymbol{c}}
+                                                    &= \boldsymbol{z}_j\\
+                                              \boldsymbol{z}_j
+                                                    &\in \mathcal{P}_{d_j}
+                                          \end{array}
+                                          \hspace{5mm} \forall j\in\mathcal{J}.
+    \end{aligned}
+    \label{eq:lp:admm_reformulated}
+\end{align}
+%
+In this form, the problem almost fits the \ac{ADMM} template described in section
+\ref{sec:theo:Optimization Methods}, except for the fact that there are multiple equality
+constraints $\boldsymbol{T}_j \tilde{\boldsymbol{c}} = \boldsymbol{z}_j$ and the
+additional constraints $\boldsymbol{z}_j \in \mathcal{P}_{d_j} \, \forall\, j\in\mathcal{J}$.
+\todo{$\forall$ in text?}
+The multiple constraints can be addressed by introducing additional terms in the
+augmented lagrangian:%
+%
+\begin{align*}
+    \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \boldsymbol{z}_{1:m},
+        \boldsymbol{\lambda}_{1:m} \right) 
+    = \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
+        + \sum_{j\in\mathcal{J}} \boldsymbol{\lambda}^\text{T}_j
+        \left( \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j \right) 
+        + \frac{\mu}{2}\sum_{j\in\mathcal{J}}
+            \lVert \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j \rVert^2_2
+.\end{align*}%
+%
+The additional constraints remain in the dual optimization problem:%
+%
+\begin{align*}
+    \text{maximize } \min_{\substack{\tilde{\boldsymbol{c}} \\
+            \boldsymbol{z}_j \in \mathcal{P}_{d_j}}}
+    \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \boldsymbol{z}_{1:m},
+        \boldsymbol{\lambda}_{1:m} \right)
+.\end{align*}%
+%
+The steps to solve the dual problem then become:
+%
+\begin{alignat*}{3}
+    \tilde{\boldsymbol{c}} &\leftarrow \argmin_{\tilde{\boldsymbol{c}}} \mathcal{L}_{\mu} \left( 
+        \tilde{\boldsymbol{c}}, \boldsymbol{z}_{1:m}, \boldsymbol{\lambda}_{1:m} \right) \\
+    \boldsymbol{z}_j &\leftarrow \argmin_{\boldsymbol{z}_j \in \mathcal{P}_{d_j}}
+        \mathcal{L}_{\mu} \left( 
+            \tilde{\boldsymbol{c}}, \boldsymbol{z}_{1:m}, \boldsymbol{\lambda}_{1:m} \right)
+        \hspace{3mm} &&\forall j\in\mathcal{J} \\
+    \boldsymbol{\lambda}_j &\leftarrow \boldsymbol{\lambda}_j
+        + \mu\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
+        - \boldsymbol{z}_j \right)
+        \hspace{3mm} &&\forall j\in\mathcal{J}
+.\end{alignat*}
+%
+Luckily, the additional constaints only affect the $\boldsymbol{z}_j$-update steps.
+Furthermore, the $\boldsymbol{z}_j$-update steps can be shown to be equivalent to projections
+onto the check polytopes $\mathcal{P}_{d_j}$ \cite[Sec. III. B.]{original_admm}
+and the $\tilde{\boldsymbol{c}}$-update can be computed analytically \cite[Sec. III.]{lautern}:%
+%
+\begin{alignat*}{3}
+    \tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( 
+        \sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{\lambda}_j \right)_i
+            - \left( \boldsymbol{z}_j \right)_i \Big) - \frac{\gamma_i}{\mu} \right)
+    \hspace{3mm} && \forall i\in\mathcal{I} \\
+    \boldsymbol{z}_j &\leftarrow \Pi_{\mathcal{P}_{d_j}}\left(
+        \boldsymbol{T}_j\tilde{\boldsymbol{c}} + \boldsymbol{\lambda}_j \right)
+    \hspace{3mm} && \forall j\in\mathcal{J} \\
+    \boldsymbol{\lambda}_j &\leftarrow \boldsymbol{\lambda}_j
+        + \mu\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
+        - \boldsymbol{z}_j \right)
+    \hspace{3mm} && \forall j\in\mathcal{J}
+.\end{alignat*}
+%
+One thing to note is that all of the $\boldsymbol{z}_j$-updates can be computed simultaneously,
+as they are independent of one another.
+The same is true for the updates of the individual components of $\tilde{\boldsymbol{c}}$.
+
+The reason \ac{ADMM} is able to perform so well is due to the relocation of the constraints
+$\boldsymbol{T}_j\tilde{\boldsymbol{c}}_j\in\mathcal{P}_{d_j}\,\forall\, j\in\mathcal{J}$
+into the objective function itself.
+The minimization of the new objective function can then take place simultaneously
+with respect to all $\boldsymbol{z}_j, j\in\mathcal{J}$.
+Effectively, all of the $\left|\mathcal{J}\right|$ parity constraints are
+able to be handled at the same time.
+This can also be understood by interpreting the decoding process as a message-passing
+algorithm \cite[Sec. III. D.]{original_admm}, \cite[Sec. II. B.]{efficient_lp_dec_admm},
+as is shown in figure \ref{fig:lp:message_passing} 
+\footnote{$\epsilon_{\text{pri}} > 0$ and $\epsilon_{\text{dual}} > 0$ are additional parameters
+defining the tolerances for the stopping criteria of the algorithm.
+$\boldsymbol{z}_j^\prime$ denotes the value of $\boldsymbol{z}_j$ in the previous iteration.}%
+\todo{Move footnote to figure caption}%
+.%
+\todo{Explicitly specify sections?}%
+%
+\begin{figure}[H]
+    \centering
+   
+    \begin{genericAlgorithm}[caption={}, label={}]
+Initialize $\tilde{\boldsymbol{c}}, \boldsymbol{z}_{1:m}$ and $\boldsymbol{\lambda}_{1:m}$
+while $\sum_{j\in\mathcal{J}} \lVert \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j \rVert_2 \ge \epsilon_{\text{pri}}$ or $\sum_{j\in\mathcal{J}} \lVert \boldsymbol{z}^\prime_j - \boldsymbol{z}_j \rVert_2 \ge \epsilon_{\text{dual}}$
+    Perform check update
+        ...
+    Perform variable update
+        ...
+    \end{genericAlgorithm}
+
+    \caption{\ac{LP} decoding using \ac{ADMM} interpreted as a message passing algorithm}
+    \label{fig:lp:message_passing}
+\end{figure}%
+%
+\noindent The $\tilde{c}_i$-updates can be interpreted as a variable-node update step,
+and the $\boldsymbol{z}_j$- and $\boldsymbol{\lambda}_j$-updates can be interpreted as
+a check-node update step.
+The updates for each variable- and check-node can be perfomed in parallel.
+With this interpretation it becomes clear why \ac{LP} decoding using \ac{ADMM}
+is able to achieve similar computational complexity to \ac{BP}.
+
+The main computational effort in solving the linear program then amounts to
+computing the projection operation $\Pi_{\mathcal{P}_{d_j}} \left( \cdot \right) $
+onto each check polytope. Various different methods to perform this projection
+have been proposed (e.g., in \cite{original_admm}, \cite{efficient_lp_dec_admm},
+\cite{lautern}).
+The method chosen here is the one presented in \cite{lautern}.
+
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -746,7 +885,7 @@ as is often done, and has a rather unwieldy representation:%
     f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right) =
         \frac{1}{\left| \mathcal{C} \right| }
             \sum_{\boldsymbol{c} \in \mathcal{C} }
-                \delta\left( \tilde{\boldsymbol{x}} - \left( -1 \right) ^{\boldsymbol{c}}\right)
+                \delta\big( \tilde{\boldsymbol{x}} - \left( -1 \right) ^{\boldsymbol{c}}\big)
     \label{eq:prox:prior_pdf}
 .\end{align}%
 %
@@ -758,7 +897,7 @@ the so-called \textit{code-constraint polynomial} is introduced as:%
     h\left( \tilde{\boldsymbol{x}} \right) =
         \underbrace{\sum_{i=1}^{n} \left( \tilde{x_i}^2-1 \right) ^2}_{\text{Bipolar constraint}}
         + \underbrace{\sum_{j=1}^{m} \left[
-            \left( \prod_{i\in N \left( j \right) } \tilde{x_i} \right)
+            \left( \prod_{i\in N_c \left( j \right) } \tilde{x_i} \right)
         -1 \right] ^2}_{\text{Parity constraint}}%
 .\end{align*}%
 %
@@ -795,6 +934,10 @@ $L \left( \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \right) = -\ln\left(
         + \gamma h\left( \tilde{\boldsymbol{x}} \right) 
         \right)%
 .\end{align*}%
+\todo{\textbackslash left($\cdot$ \textbackslash right)\\
+$\rightarrow$\\
+\textbackslash big( $\cdot$ \textbackslash big)\\
+?}%
 %
 Thus, with proximal decoding, the objective function
 $g\left( \tilde{\boldsymbol{x}} \right)$ considered is%
@@ -847,6 +990,7 @@ It is then immediately approximated with gradient-descent:%
     \hspace{5mm} \gamma > 0, \text{ small}
 .\end{align*}%
 %
+\todo{explicitly state $\nabla h$?}
 The second step thus becomes%
 %
 \begin{align*}