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Start VN and CN indexing from zero

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This commit is contained in:
Andreas Tsouchlos
2026-05-01 17:30:14 +02:00
parent 635c0aab18
commit 3b7618e1d1
3 changed files with 20 additions and 22 deletions
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26
src/thesis/chapters/2_fundamentals.tex
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@@ -224,25 +224,25 @@ construction for the [7,4,3]-Hamming code.
} }
\begin{tikzpicture} \begin{tikzpicture}
\node[VN, label=above:$x_1$] (vn1) {}; \node[VN, label=above:$x_0$] (vn1) {};
\node[VN, right=12mm of vn1, label=above:$x_2$] (vn2) {}; \node[VN, right=12mm of vn1, label=above:$x_1$] (vn2) {};
\node[VN, right=12mm of vn2, label=above:$x_3$] (vn3) {}; \node[VN, right=12mm of vn2, label=above:$x_2$] (vn3) {};
\node[VN, right=12mm of vn3, label=above:$x_4$] (vn4) {}; \node[VN, right=12mm of vn3, label=above:$x_3$] (vn4) {};
\node[VN, right=12mm of vn4, label=above:$x_5$] (vn5) {}; \node[VN, right=12mm of vn4, label=above:$x_4$] (vn5) {};
\node[VN, right=12mm of vn5, label=above:$x_6$] (vn6) {}; \node[VN, right=12mm of vn5, label=above:$x_5$] (vn6) {};
\node[VN, right=12mm of vn6, label=above:$x_7$] (vn7) {}; \node[VN, right=12mm of vn6, label=above:$x_6$] (vn7) {};
\node[ \node[
CN, below=25mm of vn4, CN, below=25mm of vn4,
label={below:$x_1 + x_3 + x_4 + x_6 = 0$} label={below:$x_0 + x_2 + x_3 + x_5 = 0$}
] (cn2) {}; ] (cn2) {};
\node[ \node[
CN, left=40mm of cn2, CN, left=40mm of cn2,
label={below:$x_2 + x_3 + x_4 + x_5 = 0$} label={below:$x_1 + x_2 + x_3 + x_4 = 0$}
] (cn1) {}; ] (cn1) {};
\node[ \node[
CN, right=40mm of cn2, CN, right=40mm of cn2,
label={below:$x_1 + x_2 + x_4 + x_7 = 0$} label={below:$x_0 + x_1 + x_3 + x_6 = 0$}
] (cn3) {}; ] (cn3) {};
\foreach \n in {2,3,4,5} { \foreach \n in {2,3,4,5} {
@@ -268,9 +268,9 @@ construction for the [7,4,3]-Hamming code.
% %
Mathematically, we represent a \ac{vn} using the index $i \in Mathematically, we represent a \ac{vn} using the index $i \in
\mathcal{I} := \left[ \mathcal{I} := \left[ 0:n-1 \right] := \left\{ 0,1,\ldots,n-1 \right\}$
1 : n \right]$ and a \ac{cn} using the index $j \in \mathcal{J} and a \ac{cn} using the index $j \in \mathcal{J}
:= \left[ 1 : m \right]$. := \left[ 0 : m-1 \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 variable node $i$ as the neighborhood of a variable node $i$ as
$\mathcal{N}_\text{V} (i) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i} $\mathcal{N}_\text{V} (i) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i}

6
src/thesis/chapters/3_fault_tolerant_qec.tex
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@@ -41,7 +41,7 @@ address both.
% Definition of fault tolerance % Definition of fault tolerance
We model the possible occurrence of errors during any processing We model the possible occurrence of errors during any processing
stage as different \emph{error locations} $E_i,~i\in \{1,\ldots,N\}$ stage as different \emph{error locations} $E_i,~i\in [1:N]$
in the circuit. in the circuit.
$N \in \mathbb{N}$ is the total number of considered error locations. $N \in \mathbb{N}$ is the total number of considered error locations.
The \emph{circuit error vector} $\bm{e} \in \{0,1\}^N$ is a vector The \emph{circuit error vector} $\bm{e} \in \{0,1\}^N$ is a vector
@@ -861,7 +861,7 @@ can add the results from the previous round, as illustrated in
\Cref{fig:detectors_from_measurements_general}. \Cref{fig:detectors_from_measurements_general}.
We thus have $D=n-k$. We thus have $D=n-k$.
Concretely, we denote the outcome of Concretely, we denote the outcome of
measurement $\ell \in \{1,\ldots,n-k\}$ in round $r \in \{1,\ldots,R\}$ by measurement $\ell \in [1:n-k]$ in round $r \in [1:R]$ by
$m_\ell^{(r)} \in \mathbb{F}_2$ $m_\ell^{(r)} \in \mathbb{F}_2$
and define and define
\begin{gather*} \begin{gather*}
@@ -873,7 +873,7 @@ and define
\end{pmatrix} \end{pmatrix}
.% .%
\end{gather*} \end{gather*}
Similarly, we denote the outcome of detector $j\in\{1,\ldots,D\}$ in Similarly, we denote the outcome of detector $j\in[1:D]$ in
round $r$ by $d_j^{(r)} \in \mathbb{F}_2$ and define round $r$ by $d_j^{(r)} \in \mathbb{F}_2$ and define
\begin{gather} \begin{gather}
\label{eq:measurement_combination} \label{eq:measurement_combination}

10
src/thesis/chapters/4_decoding_under_dems.tex
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@@ -474,9 +474,8 @@ and is difficult to predict beforehand.
We briefly reintroduce the notation important for the definition of the windows. We briefly reintroduce the notation important for the definition of the windows.
We use the variables $n,m \in \mathbb{N}$ to describe the number of We use the variables $n,m \in \mathbb{N}$ to describe the number of
\acp{vn} and \acp{cn} respectively. \acp{vn} and \acp{cn} respectively.
We index the \acp{vn} using the variable $i \in \mathcal{I} := \left\{ We index the \acp{vn} using the variable $i \in \mathcal{I} :=
1,\ldots,n \right\}$ and the \acp{cn} using the variable $j \in [0:n-1]$ and the \acp{cn} using the variable $j \in \mathcal{J} := [ 0 : m-1]$.
\mathcal{J} := \left\{ 1, \ldots, m \right\}$.
Finally, we call $\mathcal{N}_\text{V}(i) = \left\{ i\in \mathcal{I}: Finally, we call $\mathcal{N}_\text{V}(i) = \left\{ i\in \mathcal{I}:
\bm{H}_{j,i} = 1 \right\}$ and $\mathcal{N}_\text{C}(j) := \left\{ j \bm{H}_{j,i} = 1 \right\}$ and $\mathcal{N}_\text{C}(j) := \left\{ j
\in \mathcal{J} : \bm{H}_{j,i} = 1 \right\}$ the neighborhoods of the \in \mathcal{J} : \bm{H}_{j,i} = 1 \right\}$ the neighborhoods of the
@@ -487,9 +486,8 @@ check matrix of the underlying code, from which the \ac{dem} was generated.
% How we get the corresponding rows % How we get the corresponding rows
We begin by describing the sets of \acp{cn} relevant to each window. We begin by describing the sets of \acp{cn} relevant to each window.
For indexing, we use the variable $\ell \in \left\{ For indexing, we use the variable $\ell \in [0:n_\text{win} - 1]$,
0,\ldots,n_\text{win} - 1 \right\}$, where $n_\text{win} \in \mathbb{N}$ where $n_\text{win} \in \mathbb{N}$ is the number of windows.
is the number of windows.
Because we defined the step size $F$ as the number of syndrome Because we defined the step size $F$ as the number of syndrome
extraction rounds to skip, the first \ac{cn} of window $\ell$ should have index extraction rounds to skip, the first \ac{cn} of window $\ell$ should have index
$\ell F m$. $\ell F m$.