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Update bibliography, phrasing, add outlines for sections

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Andreas Tsouchlos
2026-03-27 00:15:39 +01:00
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src/thesis/chapters/2_fundamentals.tex
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@@ -21,7 +21,8 @@ reduced error rate.
Specifically, Shannon proved in 1948 that for any channel, a block
code can be found that achieves arbitrarily small probability of
error at any communication rate up to the capacity of the channel
when the block length approaches infinity \cite{shannon_mathematical_1948}.
when the block length approaches infinity
\cite[Sec.~13]{shannon_mathematical_1948}.
In this section, we explore the concepts of ``classical'' (as in non-quantum)
error correction that are central to this work.
@@ -52,7 +53,7 @@ $\bm{u} \in \mathbb{F}_2^k$ of length $k \in \mathbb{N}$ (called the
A measure of the amount of introduced redundancy is the \textit{code
rate} $R = k/n$.
We call the set of all codewords $\mathcal{C}$ the \textit{code}
\cite[Sec. 3.1]{ryan_channel_2009}.
\cite[Sec.~3.1.1]{ryan_channel_2009}.
%
% d_min and the [] Notation
@@ -69,14 +70,14 @@ $\bm{x}_2$ can be expressed using the \textit{Hamming distance} $d(\bm{x}_1,
We define the \textit{minimum distance} of a code $\mathcal{C}$ as
%
\begin{align*}
d_\text{min} = \min \left\{ d(\bm{x}_1, \bm{x}_2) : \bm{x}_1,
d_\text{min} := \min \left\{ d(\bm{x}_1, \bm{x}_2) : \bm{x}_1,
\bm{x}_2 \in \mathcal{C}, \bm{x}_1 \neq \bm{x}_2 \right\}
.
\end{align*}
%
We can signify that a binary linear block code has information length
$k$, block length $n$ and minimum distance $d_\text{min}$ using the
notation $[n,k,d_\text{dmin}]$ \cite[Sec. 1.3]{macwilliams_theory_1977}.
notation $[n,k,d_\text{dmin}]$ \cite[Sec.~1.3]{macwilliams_theory_1977}.
%
% Parity checks, H, and the syndrome
@@ -90,16 +91,19 @@ Since $\lvert \mathcal{C} \rvert = 2^k$ and $\lvert \mathbb{F}_2^n
additional degrees of freedom.
These conditions, called parity checks, take the form of equations
over $\mathbb{F}_2^n$, linking the individual positions of each codeword.
We can arrange the coefficients of these equations in the
We can arrange the coefficients of these equations in a
\textit{parity-check matrix} (\acs{pcm}) $\bm{H} \in
\mathbb{F}_2^{(n-k) \times n}$ and equivalently define the code as
\cite[Sec. 3.1]{ryan_channel_2009}
\cite[Sec.~3.1.1]{ryan_channel_2009}
%
\begin{align*}
\mathcal{C} = \left\{ \bm{x} \in \mathbb{F}_2^n :
\bm{H}\bm{x}^\text{T} = \bm{0} \right\}
.%
\end{align*}
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
@@ -113,15 +117,14 @@ exponentially with $n$, in contrast to keeping track of all codewords directly.
%
Figure \ref{fig:Diagram of a transmission system} visualizes the
entire communication process \cite[Sec. 1.1]{ryan_channel_2009}.
communication process \cite[Sec.~1.1]{ryan_channel_2009}.
An input message $\bm{u}\in \mathbb{F}_2^k$ is mapped onto a codeword $\bm{x}
\in \mathbb{F}_2^n$. This is passed on to a modulator, which
interacts with the physical channel.
A demodulator processes the received message and forwards the result
A demodulator processes the channel output and forwards the result
$\bm{y} \in \mathbb{R}^n$ to a decoder.
Finally, the decoder is responsible for obtaining an estimate
$\hat{\bm{u}} \in \mathbb{F}_2^k$ of the original input message from the
received message.
$\hat{\bm{u}} \in \mathbb{F}_2^k$ of the original input message.
This is done by first finding an estimate $\hat{\bm{x}}$ of the sent
codeword and undoing the encoding.
The decoding problem that we generally attempt to solve thus consists
@@ -133,9 +136,9 @@ One approach is to use the \ac{ml} criterion \cite[Sec.
P(\bm{Y} = \bm{y} \vert \bm{X} = \bm{x})
.
\end{align*}
Finally, we differentiate between \textit{soft decision} decoding, where
$\bm{y} \in \mathbb{R}^n$, and \textit{hard decision} decoding, where
$\bm{y} \in \mathbb{F}_2^n$ \cite[Sec. 1.5.1.3]{ryan_channel_2009}.
Finally, we differentiate between \textit{soft-decision} decoding, where
$\bm{y} \in \mathbb{R}^n$, and \textit{hard-decision} decoding, where
$\bm{y} \in \mathbb{F}_2^n$ \cite[Sec.~1.5.1.3]{ryan_channel_2009}.
%
\begin{figure}[h]
\centering
@@ -199,7 +202,7 @@ This is exactly the motivation behind \ac{ldpc} codes \cite[Ch.
%
\ac{ldpc} codes belong to a class sometimes referred to as ``modern codes''.
These differ from ``classical codes'' in their decoding algorithm:
These differ from ``classical codes'' in their decoding algorithms:
Classical codes are usually decoded using one-step hard-decision decoding,
whereas modern codes are suitable for iterative soft-decision
decoding \cite[Preface]{ryan_channel_2009}. The iterative decoding algorithms
@@ -209,8 +212,8 @@ graph that constitues an alternative representation of the \ac{pcm}.
We define two types of nodes: \acp{vn}, corresponding to codeword
bits, and \acp{cn}, corresponding to individual parity checks.
We then construct the Tanner graph by connecting each \ac{cn} to
the \acp{vn} that make up the corresponding parity check \cite[Ch.
5]{ryan_channel_2009}.
the \acp{vn} that make up the corresponding parity check
\cite[Sec.~5.1.2]{ryan_channel_2009}.
Figure \ref{PCM and Tanner graph of the Hamming code} shows this
construction for the [7,4,3]-Hamming code.
%
@@ -290,30 +293,46 @@ the neighborhood of a varialbe node $i$ as
$\mathcal{N}_\text{V} (i) = \left\{ i \in \mathcal{I} : \bm{H}_{j,i}
= 1 \right\}$
and that of a check node $j$ as
$\mathcal{N}_\text{C} = \left\{ j \in \mathcal{J} : \bm{H}_{j,i} = 1 \right\}$.
$\mathcal{N}_\text{C} (j) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i}
= 1 \right\}$.
\red{
\begin{itemize}
\item Cycles (? - Only if needed later)
\item Regular vs irregular (? - only if needed later)
\end{itemize}
}
% TODO: Do we need any of these?
% \red{
% \begin{itemize}
% \item Cycles (? - Only if needed later)
% \item Regular vs irregular (? - only if needed later)
% \end{itemize}
% }
\subsection{Spatially-Coupled LDPC Codes}
A relatively recent development in the world of \ac{ldpc} codes is
that of \ac{sc}-\ac{ldpc} codes.\\
\red{[a bit more history (developed by \ldots, developed from \ldots,
\ldots)]}\\
\red{[core concept]}
\red{
\begin{itemize}
\item Core idea
\item Mathematical description (H)
\item Tanner graph + PCM
\item Key benefits and reasoning behind them
\item Cite \cite{costello_spatially_2014} \cite{hassan_fully_2016}
\end{itemize}
}
\subsection{Belief Propagation}
\red{[short intro]} \\
\red{[key points (sub-optimal but good enough, low complexity, \ldots)]} \\
\red{[top-level overview (iterative algorithm that approximates \ldots)]}
\red{
\begin{itemize}
\item Core idea
\item BP for SC-LDPC codes
\item SPA and NMS algorithms
% TODO: Would it be better to split this into a separate section?
\item Sliding-window decoding of SC-LDPC codes
\item Cite \cite{ryan_channel_2009} \cite{hassan_fully_2016}
\cite{costello_spatially_2014}
\end{itemize}
}