diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex
index 8d4d60d..429af0e 100644
--- a/latex/thesis/chapters/decoding_techniques.tex
+++ b/latex/thesis/chapters/decoding_techniques.tex
@@ -142,6 +142,7 @@ which minimizes the objective function $f$ (as shown in figure \ref{fig:dec:spac
 \label{sec:dec:LP Decoding using ADMM}
 
 \Ac{LP} decoding is a subject area introduced by Feldman et al.
+\todo{Space before citation?}
 \cite{feldman_paper}. They reframed the decoding problem as an
 \textit{integer linear program} and subsequently presented a relaxation into
 a \textit{linear program}, lifting the integer requirement.
@@ -152,27 +153,22 @@ work is the \ac{ADMM}.
 
 Feldman at al. begin by looking at the \ac{ML} decoding problem%
 \footnote{They assume that all codewords are equally likely to be transmitted,
-making the \ac{ML} and \ac{MAP} decoding problems essentially equivalent}%
-\todo{Dot after footnote?}%
+making the \ac{ML} and \ac{MAP} decoding problems essentially equivalent.}%
 %
 \begin{align*}
-    \hat{\boldsymbol{x}} = \argmax_{\boldsymbol{x} \in
-        \left\{ \left( -1 \right)^{\boldsymbol{c}}
-            \text{ : } \boldsymbol{c} \in \mathcal{C} \right\} }
-        f_{\boldsymbol{Y} \mid \boldsymbol{X}} \left( \boldsymbol{y} \mid \boldsymbol{x} \right) 
+    \hat{\boldsymbol{c}} = \argmax_{\boldsymbol{c} \in \mathcal{C}}
+        f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} \right) 
 .\end{align*}
 %
-\todo{Define $\mathcal{X}$ as $\left\{ \left( -1 \right)
-    ^{\boldsymbol{c}} : \boldsymbol{c}\in \mathcal{C} \right\} $?}%
 They suggest that maximizing the likelihood
-$f_{\boldsymbol{Y} \mid \boldsymbol{X}}\left( \boldsymbol{y} \mid \boldsymbol{x} \right)$
+$f_{\boldsymbol{Y} \mid \boldsymbol{C}}\left( \boldsymbol{y} \mid \boldsymbol{c} \right)$
 is equivalent to minimizing the negative log-likelihood.
 
-\ldots
+\ldots (Explaing arriving at cost function from ML decoding problem)
 
 Based on this, they propose their cost function%
 \footnote{In this context, \textit{cost function} and \textit{objective function}
-mean the same thing}
+mean the same thing.}
 for the \ac{LP} decoding problem:%
 %
 \begin{align*}
@@ -184,8 +180,29 @@ for the \ac{LP} decoding problem:%
                 \left( Y_i = y_i | C_i = 1 \right) } \right) \\
 .\end{align*}
 %
+%
+The exact integer linear program \todo{ILP acronym?} formulation of \ac{ML}
+decoding is the following:%
+%
+\begin{align*}
+    \text{minimize }\hspace{2mm} &\sum_{i=1}^{n} \gamma_i c_i \\
+    \text{subject to }\hspace{2mm} &\boldsymbol{c} \in \mathcal{C}
+.\end{align*}%
+%
 
-The 
+\ldots (LP Relaxation)
+
+%They go on to define the constraints under which this minimization is to be
+%accomplished.
+%They define the concept of the \textit{codeword polytope} as a linear
+%combination of all possible codewords, forming their convex hull:%
+%%
+%\begin{align*}
+%    \text{poly}\left( \mathcal{C} \right) = \left\{
+%        \sum_{c \in \mathcal{C}} \lambda_{\boldsymbol{c}} \boldsymbol{c}
+%            \text{ : } \lambda_{\boldsymbol{c}} \ge 0,
+%        \sum_{\boldsymbol{c} \in \mathcal{C}} \lambda_{\boldsymbol{c}} = 1 \right\} 
+%.\end{align*}
 
 \begin{itemize}
     \item Equivalent \ac{ML} optimization problem
diff --git a/latex/thesis/thesis.tex b/latex/thesis/thesis.tex
index 95869f0..1021316 100644
--- a/latex/thesis/thesis.tex
+++ b/latex/thesis/thesis.tex
@@ -179,7 +179,29 @@
     % 7. Conclusion
     %   - Summary of results
     %   - Future work
-    
+   
+
+    % Proposed new structure:
+    %
+    % 1. Introduction
+    %
+    % 2. Theoretical Background
+    %   \ldots
+    %
+    % 3. Proximal Decoding
+    %   3.1 Theory
+    %   3.2 Implementation details
+    %   3.3 Results
+    %   3.x Improved implementation
+    %
+    % 4. LP Decoding using ADMM
+    %   4.1 Theory
+    %   4.2 Implementation details
+    %   4.3 Results and comparison with proximal
+    %
+    % 5. Discussion
+    %
+    % 6. Conclusion
 
 
     \tableofcontents