Write ldpc section

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}
 }