Start VN and CN indexing from zero

src/thesis/chapters/2_fundamentals.tex

@@ -224,25 +224,25 @@ construction for the [7,4,3]-Hamming code.
     }
 
     \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[VN, label=above:$x_0$]                    (vn1) {};
+        \node[VN, right=12mm of vn1, label=above:$x_1$] (vn2) {};
+        \node[VN, right=12mm of vn2, label=above:$x_2$] (vn3) {};
+        \node[VN, right=12mm of vn3, label=above:$x_3$] (vn4) {};
+        \node[VN, right=12mm of vn4, label=above:$x_4$] (vn5) {};
+        \node[VN, right=12mm of vn5, label=above:$x_5$] (vn6) {};
+        \node[VN, right=12mm of vn6, label=above:$x_6$] (vn7) {};
 
         \node[
             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) {};
         \node[
             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) {};
         \node[
             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) {};
 
         \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 $i \in
-\mathcal{I} := \left[
-1 : n \right]$ and a \ac{cn} using the index $j \in \mathcal{J}
-:= \left[ 1 : m \right]$.
+\mathcal{I} := \left[ 0:n-1 \right] := \left\{ 0,1,\ldots,n-1 \right\}$
+and a \ac{cn} using the index $j \in \mathcal{J}
+:= \left[ 0 : m-1 \right]$.
 We can then encode the information contained in the graph by defining
 the neighborhood of a variable node $i$ as
 $\mathcal{N}_\text{V} (i) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i}

src/thesis/chapters/3_fault_tolerant_qec.tex

@@ -41,7 +41,7 @@ address both.
 % Definition of fault tolerance
 
 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.
 $N \in \mathbb{N}$ is the total number of considered error locations.
 The \emph{circuit error vector} $\bm{e} \in \{0,1\}^N$ is a vector
@@ -861,7 +861,7 @@ can add the results from the previous round, as illustrated in
 \Cref{fig:detectors_from_measurements_general}.
 We thus have $D=n-k$.
 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
 $m_\ell^{(r)} \in \mathbb{F}_2$
 and define
 \begin{gather*}
@@ -873,7 +873,7 @@ and define
     \end{pmatrix}
     .%
 \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
 \begin{gather}
     \label{eq:measurement_combination}

src/thesis/chapters/4_decoding_under_dems.tex

@@ -474,9 +474,8 @@ and is difficult to predict beforehand.
 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
 \acp{vn} and \acp{cn} respectively.
-We index the \acp{vn} using the variable $i \in \mathcal{I} := \left\{
-1,\ldots,n \right\}$ and the \acp{cn} using the variable $j \in
-\mathcal{J} := \left\{ 1, \ldots, m \right\}$.
+We index the \acp{vn} using the variable $i \in \mathcal{I} :=
+[0:n-1]$ and the \acp{cn} using the variable $j \in \mathcal{J} := [ 0 : m-1]$.
 Finally, we call $\mathcal{N}_\text{V}(i) = \left\{ i\in \mathcal{I}:
 \bm{H}_{j,i} = 1 \right\}$ and $\mathcal{N}_\text{C}(j) := \left\{ j
 \in \mathcal{J} : \bm{H}_{j,i} = 1 \right\}$ 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
 
 We begin by describing the sets of \acp{cn} relevant to each window.
-For indexing, we use the variable $\ell \in \left\{
-0,\ldots,n_\text{win} - 1 \right\}$, where $n_\text{win} \in \mathbb{N}$
-is the number of windows.
+For indexing, we use the variable $\ell \in [0:n_\text{win} - 1]$,
+where $n_\text{win} \in \mathbb{N}$ is the number of windows.
 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
 $\ell F m$.