diff --git a/src/thesis/acronyms.tex b/src/thesis/acronyms.tex index abd0fe7..8db8795 100644 --- a/src/thesis/acronyms.tex +++ b/src/thesis/acronyms.tex @@ -27,3 +27,13 @@ short=PCM, long=parity-check matrix } + +\DeclareAcronym{vn}{ + short=VN, + long=variable node +} + +\DeclareAcronym{cn}{ + short=CN, + long=chek node +} diff --git a/src/thesis/bibliography.bib b/src/thesis/bibliography.bib index 43034ef..d245eb5 100644 --- a/src/thesis/bibliography.bib +++ b/src/thesis/bibliography.bib @@ -1445,7 +1445,8 @@ We study the performance of medium-length quantum LDPC (QLDPC) codes in the depo author = {Ryan, William and Lin, Shu}, month = sep, year = {2009}, - keywords = {/unread, Computers / Networking / General, Mathematics / Applied, Technology \& Engineering / Electrical, Technology \& Engineering / Electronics / General, Technology \& Engineering / Signals \& Signal Processing, Technology \& Engineering / Telecommunications}, + keywords = {Mathematics / Applied, /unread, Technology \& Engineering / Electrical, Computers / Networking / General, Technology \& Engineering / Electronics / General, Technology \& Engineering / Signals \& Signal Processing, Technology \& Engineering / Telecommunications}, + file = {PDF:/home/andreas/workspace/work/hiwi/Zotero/storage/47TG5EWT/Ryan and Lin - 2009 - Channel Codes Classical and Modern.pdf:application/pdf}, } @book{macwilliams_theory_1977, @@ -1459,3 +1460,35 @@ We study the performance of medium-length quantum LDPC (QLDPC) codes in the depo keywords = {/unread, Computers / Information Theory}, file = {PDF:/home/andreas/workspace/work/hiwi/Zotero/storage/25RR5P4A/MacWilliams and Sloane - 1977 - The Theory of Error-correcting Codes.pdf:application/pdf}, } + +@book{richardson_modern_2008, + address = {Cambridge}, + title = {Modern {Coding} {Theory}}, + isbn = {978-0-521-85229-6}, + url = {https://www.cambridge.org/core/books/modern-coding-theory/A08C3B7B15351BF9C956CDFE5BE4846B}, + doi = {10.1017/CBO9780511791338}, + abstract = {Having trouble deciding which coding scheme to employ, how to design a new scheme, or how to improve an existing system? This summary of the state-of-the-art in iterative coding makes this decision more straightforward. With emphasis on the underlying theory, techniques to analyse and design practical iterative coding systems are presented. Using Gallager's original ensemble of LDPC codes, the basic concepts are extended for several general codes, including the practically important class of turbo codes. The simplicity of the binary erasure channel is exploited to develop analytical techniques and intuition, which are then applied to general channel models. A chapter on factor graphs helps to unify the important topics of information theory, coding and communication theory. Covering the most recent advances, this text is ideal for graduate students in electrical engineering and computer science, and practitioners. Additional resources, including instructor's solutions and figures, available online: www.cambridge.org/9780521852296.}, + urldate = {2026-03-22}, + publisher = {Cambridge University Press}, + author = {Richardson, Tom and Urbanke, Rüdiger}, + year = {2008}, + note = {TLDR: This summary of the state-of-the-art in iterative coding makes this decision more straightforward, with emphasis on the underlying theory, techniques to analyse and design practical iterative codes systems.}, + keywords = {/unread}, + file = {PDF:/home/andreas/workspace/work/hiwi/Zotero/storage/62YP4N87/Richardson and Urbanke - 2008 - Modern Coding Theory.pdf:application/pdf}, +} + +@phdthesis{gallager_low_1960, + type = {Thesis}, + title = {Low density parity check codes}, + copyright = {M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.}, + url = {https://dspace.mit.edu/handle/1721.1/11804}, + abstract = {Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering, 1960.}, + language = {eng}, + urldate = {2026-03-24}, + school = {Massachusetts Institute of Technology}, + author = {Gallager, Robert G.}, + year = {1960}, + note = {Accepted: 2005-08-17T12:00:00Z}, + keywords = {/unread}, + file = {Full Text PDF:/home/andreas/workspace/work/hiwi/Zotero/storage/PRCEXIWQ/Gallager - 1960 - Low density parity check codes.pdf:application/pdf}, +} diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index c3d4973..85d79b4 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/src/thesis/chapters/2_fundamentals.tex @@ -15,12 +15,13 @@ 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 a finite amount of redundancy, introduced -deliberately and systematically to information before its -tranmission, can be utilized to reduce the error rate of a -communications system considerably. -This idea has been expanded upon significantly since it was first -brought forward by Claude Shannon in 1948 \cite{shannon_mathematical_1948}. +realization that the introduction of a finite amount of redundancy +to information before its transmission can leed to a considerably +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}. In this section, we explore the concepts of ``classical'' (as in non-quantum) error correction that are central to this work. @@ -66,11 +67,13 @@ This notion of the distance between two codewords $\bm{x}_1$ and $\bm{x}_2$ can be expressed using the \textit{Hamming distance} $d(\bm{x}_1, \bm{x}_2)$, which is defined as the number of positions in which they differ. 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, \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}. @@ -91,11 +94,14 @@ We can arrange the coefficients of these equations in the \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} +% \begin{align*} \mathcal{C} = \left\{ \bm{x} \in \mathbb{F}_2^n : \bm{H}\bm{x}^\text{T} = \bm{0} \right\} .% \end{align*} +% TODO: Define m +% 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. The representation using the \ac{pcm} has the benefit of providing a @@ -171,11 +177,125 @@ $\bm{y} \in \mathbb{F}_2^n$ \cite[Sec. 1.5.1.3]{ryan_channel_2009}. \subsection{Low-Density Parity-Check Codes} +% +% Core concept +% + +Shannon's noisy-channel coding theorem is stated for codes whose block +length approaches infinity. This suggests that as the block length +becomes larger, the performance of the considered condes should +generally improve. +However, the size of the \ac{pcm}, and thus in general the decoding complexity, +of a linear block code grows quadratically with $n$. +This would quickly render decoding intractable as we increase the block length. +We can get around this problem by constructing $\bm{H}$ in such a +manner that the number of nonzero entries grows less than quadratically, e.g., +only linearly. +This is exactly the motivation behind \ac{ldpc} codes \cite[Ch. +1]{gallager_low_1960}. + +% +% Tanner Graph, VNs and CNs +% + +\ac{ldpc} codes belong to a class sometimes referred to as ``modern codes''. +These differ from ``classical codes'' in their decoding algorithm: +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 +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}. +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}. +Figure \ref{PCM and Tanner graph of the Hamming code} shows this +construction for the [7,4,3]-Hamming code. +% +\begin{figure}[H] + \centering + + \begin{align*} + \bm{H} = + \begin{pmatrix} + 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ + 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ + 1 & 1 & 0 & 1 & 0 & 0 & 1 \\ + \end{pmatrix} + \end{align*} + + \vspace*{2mm} + + \tikzset{ + VN/.style={ + circle, fill=KITgreen, minimum width=1mm, minimum height=1mm, + }, + CN/.style={ + rectangle, fill=KITblue, minimum width=1mm, minimum height=1mm, + }, + } + + \begin{tikzpicture} + \node[VN, label=above:$x_1$] (vn1) {}; + \node[VN, right=12mm of vn1, label=above:$x_2$] (vn2) {}; + \node[VN, right=12mm of vn2, label=above:$x_3$] (vn3) {}; + \node[VN, right=12mm of vn3, label=above:$x_4$] (vn4) {}; + \node[VN, right=12mm of vn4, label=above:$x_5$] (vn5) {}; + \node[VN, right=12mm of vn5, label=above:$x_6$] (vn6) {}; + \node[VN, right=12mm of vn6, label=above:$x_7$] (vn7) {}; + + \node[ + CN, below=25mm of vn4, + label={below:$x_1 + x_3 + x_4 + x_6 = 0$} + ] (cn2) {}; + \node[ + CN, left=40mm of cn2, + label={below:$x_2 + x_3 + x_4 + x_5 = 0$} + ] (cn1) {}; + \node[ + CN, right=40mm of cn2, + label={below:$x_1 + x_2 + x_4 + x_7 = 0$} + ] (cn3) {}; + + \foreach \n in {2,3,4,5} { + \draw (cn1) -- (vn\n); + } + + \foreach \n in {1,3,4,6} { + \draw (cn2) -- (vn\n); + } + + \foreach \n in {1,2,4,7} { + \draw (cn3) -- (vn\n); + } + \end{tikzpicture} + + \caption{The \ac{pcm} and corresponding Tanner graph of the + [7,4,3]-Hamming code.} + \label{PCM and Tanner graph of the Hamming code} +\end{figure} + +% +% N_V(j), N_C(i) +% + +Mathematically, we represent a \ac{vn} using the index $i \in +\mathcal{I} := \left[ +1 : n \right]$ 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 +$\mathcal{N}_\text{V} (i) = \left\{ i \in \mathcal{I} : \bm{H}_{j,i} += 1 \right\}$ +and that of a check node $j$ as +$\mathcal{N}_\text{C} = \left\{ j \in \mathcal{J} : \bm{H}_{j,i} = 1 \right\}$. + \red{ \begin{itemize} - \item Use \cite[Ch. 5]{ryan_channel_2009} as a reference - \item Core concept (Large $n$ with manageable complexity) - \item Tanner graphs, VNs and CNs + \item Cycles (? - Only if needed later) + \item Regular vs irregular (? - only if needed later) \end{itemize} } diff --git a/src/thesis/main.tex b/src/thesis/main.tex index 23dc82d..c2029ae 100644 --- a/src/thesis/main.tex +++ b/src/thesis/main.tex @@ -92,7 +92,7 @@ % \listoftables % \include{abbreviations} -\bibliography{lib/cel-thesis/IEEEabrv,bibliography} +\bibliography{lib/cel-thesis/IEEEabrv,src/thesis/bibliography} \end{document}