Fix N_C/N_V notation

src/thesis/chapters/2_fundamentals.tex

@@ -546,13 +546,15 @@ The \acp{vn} additionally receive messages \cite[5.4.2]{ryan_channel_2009}
 computed from the channel outputs.
 The consolidation of the information occurs in the \ac{vn} update
 \begin{align*}
-    L_{i\rightarrow j} = \tilde{L}_i + \sum_{j'\in \mathcal{N}(i)\setminus
-    j} L_{i\leftarrow j'}
+    L_{i\rightarrow j} = \tilde{L}_i + \sum_{j'\in
+        \mathcal{N}_\text{V}(i)\setminus
+    \{j\}} L_{i\leftarrow j'}
 \end{align*}
 and the \ac{cn} update
 \begin{align*}
     L_{i\leftarrow j} = 2\cdot \tanh^{-1} \left( \prod_{i'\in
-    \mathcal{N}(j)\setminus i} \tanh \frac{L_{i'\rightarrow j}}{2} \right)
+        \mathcal{N}_\text{C}(j)\setminus \{i\}} \tanh
+    \frac{L_{i'\rightarrow j}}{2} \right)
     .
 \end{align*}
 
@@ -570,9 +572,9 @@ possible cycles and are thus especially problematic.
 A simplification of the \ac{spa} is the min-sum decoder. Here, the
 \ac{cn} update is approximated as \cite[Sec.~5.5.1]{ryan_channel_2009}
 \begin{align*}
-    L_{i \leftarrow j} = \prod_{i' \in \mathcal{N}(j)\setminus i}
+    L_{i \leftarrow j} = \prod_{i' \in \mathcal{N}_\text{C}(j)\setminus \{i\}}
     \sign \left( L_{i' \rightarrow j} \right)
-    \cdot \min_{i' \in \mathcal{N}(j)\setminus i} \lvert
+    \cdot \min_{i' \in \mathcal{N}_\text{C}(j)\setminus \{i\}} \lvert
     L_{i'\rightarrow j} \rvert
     .
 \end{align*}
@@ -1568,7 +1570,7 @@ Additionally, we amend the \ac{cn} update to consider the parity
 indicated by the syndrome, calculating
 \begin{align*}
     L_{i\leftarrow j} = 2\cdot (-1)^{s_j} \cdot \tanh^{-1} \left( \prod_{i'\in
-    \mathcal{N}(j)\setminus \{i\}} \tanh \frac{L_{i'\rightarrow j}}{2} \right)
+    \mathcal{N}_\text{C}(j)\setminus \{i\}} \tanh \frac{L_{i'\rightarrow j}}{2} \right)
     .
 \end{align*}
 The resulting syndrome-based \ac{bp} algorithm is shown in

src/thesis/chapters/4_decoding_under_dems.tex

@@ -899,8 +899,8 @@ To see how we realize this in practice, we reiterate the steps of the
     \right) \\[3mm]
     \text{\ac{cn} Update (Min-Sum): }&
     \displaystyle L_{i \leftarrow j} = (-1)^{s_j}\cdot \prod_{i'
-    \in \mathcal{N}(j)\setminus \{i\}} \sign \left( L_{i' \rightarrow j}
-    \right) \cdot \min_{i' \in \mathcal{N}(j)\setminus \{i\}} \lvert
+    \in \mathcal{N}_\text{C}(j)\setminus \{i\}} \sign \left( L_{i' \rightarrow j}
+    \right) \cdot \min_{i' \in \mathcal{N}_\text{C}(j)\setminus \{i\}} \lvert
     L_{i'\rightarrow j} \rvert \\[3mm]
     \label{eq:vn_update}
     \text{\ac{vn} Update: } & \displaystyle L_{i \rightarrow j} =