Finish first draft of classical fundamentals

src/thesis/acronyms.tex

@@ -38,6 +38,11 @@
     long=maximum likelihood
 }
 
+\DeclareAcronym{map}{
+    short=MAP,
+    long=maximum a posteriori
+}
+
 \DeclareAcronym{pcm}{
     short=PCM,
     long=parity-check matrix

src/thesis/bibliography.bib

@@ -1548,3 +1548,18 @@ We study the performance of medium-length quantum LDPC (QLDPC) codes in the depo
 	pages = {168--176},
 	file = {Full Text PDF:/home/andreas/workspace/work/hiwi/Zotero/storage/WH3R5BMN/Costello et al. - 2014 - Spatially coupled sparse codes on graphs theory and practice.pdf:application/pdf},
 }
+
+@article{hagenauer_iterative_2002,
+	title = {Iterative decoding of binary block and convolutional codes},
+	volume = {42},
+	url = {https://ieeexplore.ieee.org/abstract/document/485714/},
+	number = {2},
+	urldate = {2026-04-09},
+	journal = {IEEE Transactions on information theory},
+	publisher = {IEEE},
+	author = {Hagenauer, Joachim and Offer, Elke and Papke, Lutz},
+	year = {2002},
+	keywords = {/unread},
+	pages = {429--445},
+	file = {Available Version (via Google Scholar):/home/andreas/workspace/work/hiwi/Zotero/storage/BYT9IHNL/Hagenauer et al. - 2002 - Iterative decoding of binary block and convolutional codes.pdf:application/pdf},
+}

src/thesis/chapters/2_fundamentals.tex

@@ -404,13 +404,14 @@ 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.
 This is achieved by connecting some \acp{vn} of one spatial position to
 \acp{cn} of another, resulting in a \ac{pcm} of the form
 \cite[Eq.~1]{hassan_fully_2016}
-% TODO: Find reference and make sure notation is correct
 %
 \begin{align*}
     \bm{H} =
@@ -429,7 +430,6 @@ where $W \in \mathbb{N}$ is the \textit{coupling width} and $L \in
 This construction results in a Tanner graph as depicted in
 \autoref{fig:sc-ldpc-tanner}.
 
-% TODO: Create SC-LDPC graphic
 \begin{figure}[t]
     \centering
 
@@ -442,7 +442,6 @@ This construction results in a Tanner graph as depicted in
         },
     }
 
-    % TODO: Coupling over more spatial positions
     \begin{tikzpicture}[node distance=7mm and 1cm]
         \node[VN]                  (vn00) {};
         \node[VN, below = of vn00] (vn01) {};
@@ -523,41 +522,89 @@ This construction results in a Tanner graph as depicted in
 Note that at the first and last few spatial positions, some \acp{cn}
 have lower degrees.
 This leads to more reliable information about the
-\acp{vn} at those positions, that, as we will see, is
+\acp{vn} that, as we will see, is
 later passed to subsequent spatial positions during decoding.
 This is precisely the effect that leads to the good performance of
 \ac{sc}-\ac{ldpc} codes in the waterfall region \cite{costello_spatially_2014}.
 
 \subsection{Iterative Decoding}
 
-% TODO: Add exact reference
+% Introduction
+
 \ac{ldpc} codes are generally decoded using efficient iterative
 algorithms, something that is possilbe due to their sparsity
 \cite[Sec.~5.3]{ryan_channel_2009}.
-The algorithm originally proposed for this purpose by Gallager in
-1960 is now known as the \ac{spa} \cite[5.4.1]{ryan_channel_2009},
-also called \ac{bp}.
+The algorithm originally proposed alongside LDPC codes for this
+purpose by Gallager in 1960 is now known as the \ac{spa}
+\cite[5.4.1]{ryan_channel_2009}, also called \ac{bp}.
 
-% - SPA uses symbol-wise MAP as decision criterion for each symmbol
-% - Optimal when Tanner graph is a tree, suboptimal with cycles
-% - Use of LLRs instead of probabilties directly
-% - Actual algorithm
-% - CNs: single parity-check codes; VNs: repetition codes
-% - Algorithm
+The optimality criterion the \ac{spa} is built around is a
+symbol-wise \ac{map} decision \cite[Sec.~5.4.1]{ryan_channel_2009}.
+The core idea of the resulting algorithm is to view \acp{cn} as
+representing single-parity check codes and \acp{vn} as representing
+repetition codes.
+The algorithm alternates between consolidating soft information about
+the \acp{vn} in the \acp{cn}, and consolidating soft information abou
+the \acp{cn} in the \acp{vn}.
+To this end, messages are passed back and forth along the edges of
+the Tanner graph.
+$L_{i\rightarrow j}$ represents a message passed from \ac{vn} $i$ to
+\ac{cn} j, $L_{i\leftarrow j}$ represents a message passed from
+\ac{cn} j to \ac{vn} i.
+The \acp{vn} additionally receive messages \cite[5.4.2]{ryan_channel_2009}
+\begin{align*}
+    \tilde{L}_i = \log \frac{P(X=0 \vert Y=y)}{P(X=1 \vert Y=y)},
+\end{align*}
+computed from the channel outputs.
+The consolidation of the information occurs in the \ac{vn} update
+\begin{align*}
+    L_{i\rightarrow j} = \tilde{L}_i + \sum_{j'\in \mathcal{N}(i)\setminus
+    j} L_{i\leftarrow j'}
+\end{align*}
+and the \ac{cn} update
+\begin{align*}
+    L_{i\leftarrow j} = 2\cdot \tanh^{-1} \left( \prod_{i'\in
+    \mathcal{N}(j)\setminus i} \tanh \frac{L_{i'\rightarrow j}}{2} \right)
+    .
+\end{align*}
+
+A basic assumption for the derivation of the \ac{spa} is that the
+messages are statistically independent.
+If the Tanner graph has cycles, however, this
+condition is not met.
+The shorter the cycles, the sooner this condition is violated and the
+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
-%     Approximation of CN update by min sum operation
 
-For \ac{sc}-\ac{ldpc} codes, the iterative decoding procedure is wrapped by a
-windowing step. This is done to reduce the latency and memory and
+A simplification of the \ac{spa} is the min-sum decoder. Here, the
+\ac{cn} update is approximated as \cite[Sec.~5.5.1]{ryan_channel_2009}
+\begin{align*}
+    L_{i \leftarrow j} = \prod_{i' \in \mathcal{N}(j)\setminus i}
+    \sign \left( L_{i' \rightarrow j} \right)
+    \cdot \min_{i' \in \mathcal{N}(j)\setminus i} \lvert
+    L_{i'\rightarrow j} \rvert
+    .
+\end{align*}
+
+% Sliding-window decoding
+
+For \ac{sc}-\ac{ldpc} codes, the iterative decoding process is wrapped by a
+windowing step. This is done to reduce the latency and memory requirements and
 also the overall computational complexity \cite{costello_spatially_2014}.
-To this end, the \ac{pcm} is split into several overlapping windows.
+To this end, the Tanner graph is split into several overlapping windows.
 During decoding, the messages that are passed along the edges of the
 graph in the overlapping regions are kept in memory and used for the
-decoding of subsequent blocks \cite[Sec.~III~C.]{costello_spatially_2014}.
-
-% BP for SC-LDPC codes
-%     Windowed decoding
+decoding of subsequent blocks \cite[Sec.~III.~C.]{hassan_fully_2016}.
 
 \section{Quantum Mechanics and Quantum Information Science}
 \label{sec:Quantum Mechanics and Quantum Information Science}