diff --git a/latex/thesis/chapters/decoding_techniques.tex b/latex/thesis/chapters/decoding_techniques.tex
index ac26881..3250fdc 100644
--- a/latex/thesis/chapters/decoding_techniques.tex
+++ b/latex/thesis/chapters/decoding_techniques.tex
@@ -3,8 +3,10 @@
 
 In this chapter, the decoding techniques examined in this work are detailed.
 First, an overview of the general methodology of using optimization methods
-for channel decoding is given. Afterwards, the specific decoding techniques
-themselves are explained.
+for channel decoding is given.
+Then, the field of \ac{LP} decoding and an \ac{ADMM}-based \ac{LP} decoding
+algorithm are introduced.
+Finally, the \textit{proximal decoding} algorithm is presented.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -46,7 +48,7 @@ the viewpoint then changes from observing the decoding process in its
 tanner graph representation (as shown in figure \ref{fig:dec:tanner})
 to a spatial representation (figure \ref{fig:dec:spatial}),
 where the codewords are some of the edges of a hypercube.
-The goal is to find that point $\boldsymbol{c}$,
+The goal is to find the point $\boldsymbol{c}$,
 which minimizes the objective function $f$.
 
 %
@@ -256,12 +258,12 @@ the transfer matrix would be $\boldsymbol{T}_j =
     0 & 1 & 0 & 0 & 0 & 0 & 0 \\
     0 & 0 & 0 & 1 & 0 & 0 & 0 \\
     0 & 0 & 0 & 0 & 0 & 1 & 0 \\
-\end{bmatrix} $ (example taken from \cite[Sec. II, A]{efficient_lp_dec_admm})}
+\end{bmatrix} $ (example taken from \cite[Sec. II, A]{efficient_lp_dec_admm}).}
 (i.e. the relevant components of $\boldsymbol{c}$ for parity-check $j$)
 and $\mathcal{P}_{d}$ is the \textit{check polytope}, the convex hull of all
 binary vectors of length $d$ with even parity%
 \footnote{Essentially $\mathcal{P}_{d_j}$ is the set of vectors that satisfy
-parity-check $j$, but extended to continuous domain.}%
+parity-check $j$, but extended to the continuous domain.}%
 .
 
 In figure \ref{fig:dec:poly}, the two relaxations are compared for an