Remove TODOs

src/thesis/chapters/2_fundamentals.tex

@@ -12,8 +12,6 @@ these topics and subsequently introduces the fundamentals of \ac{qec}.
 \section{Classical Error Correction}
 \label{sec:Classical Error Correction}
 
-% TODO: Maybe rephrase: The core concept is not the realization, its's the
-% thing itself
 The core concept underpinning error correcting codes is the
 realization that introducing a finite amount of redundancy to
 information before transmission can considerably reduce the error rate.
@@ -31,14 +29,12 @@ first considering binary linear block codes in general and then \ac{ldpc} and
 Finally, we pivot to the decoding process, specifically the \ac{bp}
 algorithm.
 
-% TODO: Use subsubsections?
 \subsection{Binary Linear Block Codes}
 
 %
 % Codewords, n, k, rate
 %
 
-% TODO: Do I need a specific reference for the expanded Hilbert space thing?
 One particularly important class of coding schemes is that of binary
 linear block codes.
 The information to be protected takes the form of a sequence of
@@ -103,8 +99,6 @@ We can arrange the coefficients of these equations in a
 Note that in general we may have linearly dependent parity checks,
 prompting us to define the \ac{pcm} as $\bm{H} \in
 \mathbb{F}_2^{m\times n}$ with $\hspace{2mm} m \ge n-k$ instead.
-% TODO: Define m
-%
 The \textit{syndrome} $\bm{s} = \bm{H} \bm{v}^\text{T}$ describes
 which parity checks a candidate codeword $\bm{v} \in \mathbb{F}_2^n$ violates.
 The representation using the \ac{pcm} has the benefit of providing a
@@ -300,7 +294,6 @@ qualitative performance characteristic of an \ac{ldpc} code
 \cite[Fig.~1]{costello_spatially_2014}. We talk of the
 \textit{waterfall} and the \textit{error floor} regions.
 
-% TODO: Make this look better
 \begin{figure}[t]
     \centering
 
@@ -403,8 +396,6 @@ good error floor behavior, and capacity approaching
 iterative decoding behavior, promising good performance in the
 waterfall region \cite[Intro.]{costello_spatially_2014}.
 
-% TODO: Think of other variable for overlap lengh - W is already
-% taken as the window size
 The essential property of \ac{sc}-\ac{ldpc} codes is that codewords
 from different \textit{spatial positions}, that would ordinarily be sent
 one after the other independently, are coupled.
@@ -417,14 +408,14 @@ This is achieved by connecting some \acp{vn} of one spatial position to
     \begin{pmatrix}
         \bm{H}_0(1) &        &             \\
         \vdots      & \ddots &             \\
-        \bm{H}_W(1) &        & \bm{H}_0(L) \\
+        \bm{H}_K(1) &        & \bm{H}_0(L) \\
         & \ddots &             \\
-        &        & \bm{H}_W(L) \\
+        &        & \bm{H}_K(L) \\
     \end{pmatrix}
     ,
 \end{align*}
 %
-where $W \in \mathbb{N}$ is the \textit{coupling width} and $L \in
+where $K \in \mathbb{N}$ is the \textit{coupling width} and $L \in
 \mathbb{N}$ is the number of spatial positions.
 This construction results in a Tanner graph as depicted in
 \autoref{fig:sc-ldpc-tanner}.
@@ -508,7 +499,7 @@ This construction results in a Tanner graph as depicted in
         \draw[decorate, decoration={brace, amplitude=10pt}]
         ([xshift=-5mm,yshift=2mm]vn00.north) --
         ([xshift=5mm,yshift=2mm]vn00.north -| cn20.north)
-        node[midway, above=4mm] {W};
+        node[midway, above=4mm] {K};
     \end{tikzpicture}
 
     \caption{
@@ -576,13 +567,6 @@ worse the approximation becomes \cite[Sec.~5.4.4]{ryan_channel_2009}.
 Cycles of length four (so-called \emph{$4$-cycles}) are the shortest
 possible cycles and are thus especially problematic.
 
-% TODO: Write this pragraph
-% Cite \cite[Sec.~5.4.4]{ryan_channel_2009} (Remark 3)
-% The suboptimality of BP
-%     - Optimal when Tanner graph is a tree
-%     - The shorter the cycle, the larger the problem. 4 cycles are
-%       especially problematic
-
 % Min-sum algorithm
 
 A simplification of the \ac{spa} is the min-sum decoder. Here, the