an.tsouchlos/ma-thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 12 Wiki Activity

Remove TODOs

Browse Source
This commit is contained in:
Andreas Tsouchlos
2026-04-10 10:51:39 +02:00
parent 9edd80cf28
commit a7785f6c75
1 changed files with 4 additions and 20 deletions
Download Patch File Download Diff File Expand all files Collapse all files

24
src/thesis/chapters/2_fundamentals.tex
View File

@@ -12,8 +12,6 @@ these topics and subsequently introduces the fundamentals of \ac{qec}.
\section{Classical Error Correction} \section{Classical Error Correction}
\label{sec: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 The core concept underpinning error correcting codes is the
realization that introducing a finite amount of redundancy to realization that introducing a finite amount of redundancy to
information before transmission can considerably reduce the error rate. 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} Finally, we pivot to the decoding process, specifically the \ac{bp}
algorithm. algorithm.
% TODO: Use subsubsections?
\subsection{Binary Linear Block Codes} \subsection{Binary Linear Block Codes}
% %
% Codewords, n, k, rate % 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 One particularly important class of coding schemes is that of binary
linear block codes. linear block codes.
The information to be protected takes the form of a sequence of 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, Note that in general we may have linearly dependent parity checks,
prompting us to define the \ac{pcm} as $\bm{H} \in 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. \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 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. 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 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 \cite[Fig.~1]{costello_spatially_2014}. We talk of the
\textit{waterfall} and the \textit{error floor} regions. \textit{waterfall} and the \textit{error floor} regions.
% TODO: Make this look better
\begin{figure}[t] \begin{figure}[t]
\centering \centering
@@ -403,8 +396,6 @@ good error floor behavior, and capacity approaching
iterative decoding behavior, promising good performance in the iterative decoding behavior, promising good performance in the
waterfall region \cite[Intro.]{costello_spatially_2014}. 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 The essential property of \ac{sc}-\ac{ldpc} codes is that codewords
from different \textit{spatial positions}, that would ordinarily be sent from different \textit{spatial positions}, that would ordinarily be sent
one after the other independently, are coupled. 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} \begin{pmatrix}
\bm{H}_0(1) & & \\ \bm{H}_0(1) & & \\
\vdots & \ddots & \\ \vdots & \ddots & \\
\bm{H}_W(1) & & \bm{H}_0(L) \\ \bm{H}_K(1) & & \bm{H}_0(L) \\
& \ddots & \\ & \ddots & \\
& & \bm{H}_W(L) \\ & & \bm{H}_K(L) \\
\end{pmatrix} \end{pmatrix}
, ,
\end{align*} \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. \mathbb{N}$ is the number of spatial positions.
This construction results in a Tanner graph as depicted in This construction results in a Tanner graph as depicted in
\autoref{fig:sc-ldpc-tanner}. \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}] \draw[decorate, decoration={brace, amplitude=10pt}]
([xshift=-5mm,yshift=2mm]vn00.north) -- ([xshift=-5mm,yshift=2mm]vn00.north) --
([xshift=5mm,yshift=2mm]vn00.north -| cn20.north) ([xshift=5mm,yshift=2mm]vn00.north -| cn20.north)
node[midway, above=4mm] {W}; node[midway, above=4mm] {K};
\end{tikzpicture} \end{tikzpicture}
\caption{ \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 Cycles of length four (so-called \emph{$4$-cycles}) are the shortest
possible cycles and are thus especially problematic. 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 % Min-sum algorithm
A simplification of the \ac{spa} is the min-sum decoder. Here, the A simplification of the \ac{spa} is the min-sum decoder. Here, the