LLM review

src/thesis/acronyms.tex

@@ -55,7 +55,7 @@
 
 \DeclareAcronym{cn}{
     short=CN,
-    long=chek node
+    long=check node
 }
 
 \DeclareAcronym{ber}{

src/thesis/chapters/2_fundamentals.tex

@@ -4,7 +4,7 @@
 \Ac{qec} is a field of research combining ``classical''
 communications engineering and quantum information science.
 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 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
 % thing itself
 The core concept underpinning error correcting codes is the
-realization that the introduction of a finite amount of redundancy
-to information before its transmission can leed to a considerably
-reduced error rate.
+realization that introducing a finite amount of redundancy to
+information before transmission can considerably reduce the 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
@@ -42,7 +41,7 @@ algorithm.
 % 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 of
+The information to be protected takes the form of a sequence of
 binary symbols, which is split into separate blocks.
 Each block is encoded, transmitted, and decoded separately.
 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 $\mathbb{F}_2^k$
 onto $\mathcal{C} \subset \mathbb{F}_2^n$ takes place.
 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.
 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,
@@ -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
 $k$, 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
@@ -201,7 +200,7 @@ whereas modern codes are suitable for iterative soft-decision
 decoding \cite[Preface]{ryan_channel_2009}. The iterative decoding algorithms
 in question are generally defined in terms of message passing on the
 \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
 bits, and \acp{cn}, corresponding to individual parity checks.
 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 $j \in \mathcal{J}
 := \left[ 1 : m \right]$.
 We can then encode the information contained in the graph by defining
-the neighborhood of a varialbe node $i$ as
-$\mathcal{N}_\text{V} (i) = \left\{ i \in \mathcal{I} : \bm{H}_{j,i}
+the neighborhood of a variable node $i$ as
+$\mathcal{N}_\text{V} (i) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i}
 = 1 \right\}$
 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\}$.
 
 %
@@ -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.,
 the numbers of ones, of their rows and columns are constant
 \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
 large values \cite[Ch.~2,~Theorem~1]{gallager_low_1960},
 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,
-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}.
 
 \subsection{Spatially-Coupled LDPC Codes}
@@ -532,7 +531,7 @@ This is precisely the effect that leads to the good performance of
 % Introduction
 
 \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}.
 The algorithm originally proposed alongside LDPC codes for this
 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
 repetition codes.
 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}.
 To this end, messages are passed back and forth along the edges of
 the Tanner graph.