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LLM review

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Andreas Tsouchlos
2026-04-10 09:05:24 +02:00
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src/thesis/acronyms.tex
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@@ -55,7 +55,7 @@
\DeclareAcronym{cn}{ \DeclareAcronym{cn}{
short=CN, short=CN,
long=chek node long=check node
} }
\DeclareAcronym{ber}{ \DeclareAcronym{ber}{

29
src/thesis/chapters/2_fundamentals.tex
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@@ -4,7 +4,7 @@
\Ac{qec} is a field of research combining ``classical'' \Ac{qec} is a field of research combining ``classical''
communications engineering and quantum information science. communications engineering and quantum information science.
This chapter provides the relevant theoretical background on both of This chapter provides the relevant theoretical background on both of
these topics and subsequently introduces the the fundamentals of \ac{qec}. these topics and subsequently introduces the fundamentals of \ac{qec}.
% TODO: Is an explanation of BP with guided decimation needed in this chapter? % TODO: Is an explanation of BP with guided decimation needed in this chapter?
% TODO: Is an explanation of OSD needed chapter? % TODO: Is an explanation of OSD needed chapter?
@@ -15,9 +15,8 @@ these topics and subsequently introduces the the fundamentals of \ac{qec}.
% TODO: Maybe rephrase: The core concept is not the realization, its's the % TODO: Maybe rephrase: The core concept is not the realization, its's the
% thing itself % thing itself
The core concept underpinning error correcting codes is the The core concept underpinning error correcting codes is the
realization that the introduction of a finite amount of redundancy realization that introducing a finite amount of redundancy to
to information before its transmission can leed to a considerably information before transmission can considerably reduce the error rate.
reduced error rate.
Specifically, Shannon proved in 1948 that for any channel, a block Specifically, Shannon proved in 1948 that for any channel, a block
code can be found that achieves arbitrarily small probability of code can be found that achieves arbitrarily small probability of
error at any communication rate up to the capacity of the channel error at any communication rate up to the capacity of the channel
@@ -42,7 +41,7 @@ algorithm.
% TODO: Do I need a specific reference for the expanded Hilbert space thing? % 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 of The information to be protected takes the form of a sequence of
binary symbols, which is split into separate blocks. binary symbols, which is split into separate blocks.
Each block is encoded, transmitted, and decoded separately. Each block is encoded, transmitted, and decoded separately.
The encoding step introduces redundancy by mapping input messages The encoding step introduces redundancy by mapping input messages
@@ -62,7 +61,7 @@ We call the set of all codewords $\mathcal{C}$ the \textit{code}
During the encoding process, a mapping from $\mathbb{F}_2^k$ During the encoding process, a mapping from $\mathbb{F}_2^k$
onto $\mathcal{C} \subset \mathbb{F}_2^n$ takes place. onto $\mathcal{C} \subset \mathbb{F}_2^n$ takes place.
The input messages are mapped onto an expanded vector space, where The input messages are mapped onto an expanded vector space, where
they are ``further appart'', giving rise to the error correcting they are ``further apart'', giving rise to the error correcting
properties of the code. properties of the code.
This notion of the distance between two codewords $\bm{x}_1$ and This notion of the distance between two codewords $\bm{x}_1$ and
$\bm{x}_2$ can be expressed using the \textit{Hamming distance} $d(\bm{x}_1, $\bm{x}_2$ can be expressed using the \textit{Hamming distance} $d(\bm{x}_1,
@@ -77,7 +76,7 @@ We define the \textit{minimum distance} of a code $\mathcal{C}$ as
% %
We can signify that a binary linear block code has information length We can signify that a binary linear block code has information length
$k$, block length $n$ and minimum distance $d_\text{min}$ using the $k$, 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 $[n,k,d_\text{min}]$ \cite[Sec.~1.3]{macwilliams_theory_1977}.
% %
% Parity checks, H, and the syndrome % Parity checks, H, and the syndrome
@@ -201,7 +200,7 @@ whereas modern codes are suitable for iterative soft-decision
decoding \cite[Preface]{ryan_channel_2009}. The iterative decoding algorithms decoding \cite[Preface]{ryan_channel_2009}. The iterative decoding algorithms
in question are generally defined in terms of message passing on the in question are generally defined in terms of message passing on the
\textit{Tanner graph} of the code. The Tanner graph is a bipartite \textit{Tanner graph} of the code. The Tanner graph is a bipartite
graph that constitues an alternative representation of the \ac{pcm}. graph that constitutes an alternative representation of the \ac{pcm}.
We define two types of nodes: \acp{vn}, corresponding to codeword We define two types of nodes: \acp{vn}, corresponding to codeword
bits, and \acp{cn}, corresponding to individual parity checks. bits, and \acp{cn}, corresponding to individual parity checks.
We then construct the Tanner graph by connecting each \ac{cn} to We then construct the Tanner graph by connecting each \ac{cn} to
@@ -282,11 +281,11 @@ Mathematically, we represent a \ac{vn} using the index $i \in
1 : n \right]$ and a \ac{cn} using the index $j \in \mathcal{J} 1 : n \right]$ and a \ac{cn} using the index $j \in \mathcal{J}
:= \left[ 1 : m \right]$. := \left[ 1 : m \right]$.
We can then encode the information contained in the graph by defining We can then encode the information contained in the graph by defining
the neighborhood of a varialbe node $i$ as the neighborhood of a variable node $i$ as
$\mathcal{N}_\text{V} (i) = \left\{ i \in \mathcal{I} : \bm{H}_{j,i} $\mathcal{N}_\text{V} (i) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i}
= 1 \right\}$ = 1 \right\}$
and that of a check node $j$ as and that of a check node $j$ as
$\mathcal{N}_\text{C} (j) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i} $\mathcal{N}_\text{C} (j) = \left\{ i \in \mathcal{I} : \bm{H}_{j,i}
= 1 \right\}$. = 1 \right\}$.
% %
@@ -385,12 +384,12 @@ Broadly, there are two kinds of \ac{ldpc} codes, \textit{regular} and
Regular codes are characterized by the fact that the weights, i.e., Regular codes are characterized by the fact that the weights, i.e.,
the numbers of ones, of their rows and columns are constant the numbers of ones, of their rows and columns are constant
\cite[Sec.~5.1.1]{ryan_channel_2009}. \cite[Sec.~5.1.1]{ryan_channel_2009}.
Already during their introduction, regular \ac{ldpc} codes where shown to have Already during their introduction, regular \ac{ldpc} codes were shown to have
a minimum distance scaling linearly with the block length $n$ for a minimum distance scaling linearly with the block length $n$ for
large values \cite[Ch.~2,~Theorem~1]{gallager_low_1960}, large values \cite[Ch.~2,~Theorem~1]{gallager_low_1960},
which leads to them not exhibiting an error floor under \ac{ml} decoding. which leads to them not exhibiting an error floor under \ac{ml} decoding.
Irregular codes, on the other hand, generally do exhibit an error floor, Irregular codes, on the other hand, generally do exhibit an error floor,
their redeming quality being the ability to reach near-capacity their redeeming quality being the ability to reach near-capacity
performance in the waterfall region \cite[Intro.]{costello_spatially_2014}. performance in the waterfall region \cite[Intro.]{costello_spatially_2014}.
\subsection{Spatially-Coupled LDPC Codes} \subsection{Spatially-Coupled LDPC Codes}
@@ -532,7 +531,7 @@ This is precisely the effect that leads to the good performance of
% Introduction % Introduction
\ac{ldpc} codes are generally decoded using efficient iterative \ac{ldpc} codes are generally decoded using efficient iterative
algorithms, something that is possilbe due to their sparsity algorithms, something that is possible due to their sparsity
\cite[Sec.~5.3]{ryan_channel_2009}. \cite[Sec.~5.3]{ryan_channel_2009}.
The algorithm originally proposed alongside LDPC codes for this The algorithm originally proposed alongside LDPC codes for this
purpose by Gallager in 1960 is now known as the \ac{spa} purpose by Gallager in 1960 is now known as the \ac{spa}
@@ -544,7 +543,7 @@ The core idea of the resulting algorithm is to view \acp{cn} as
representing single-parity check codes and \acp{vn} as representing representing single-parity check codes and \acp{vn} as representing
repetition codes. repetition codes.
The algorithm alternates between consolidating soft information about The algorithm alternates between consolidating soft information about
the \acp{vn} in the \acp{cn}, and consolidating soft information abou the \acp{vn} in the \acp{cn}, and consolidating soft information about
the \acp{cn} in the \acp{vn}. the \acp{cn} in the \acp{vn}.
To this end, messages are passed back and forth along the edges of To this end, messages are passed back and forth along the edges of
the Tanner graph. the Tanner graph.