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Fix N_C/N_V notation

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This commit is contained in:
Andreas Tsouchlos
2026-05-03 14:07:04 +02:00
parent dd30b4fc0d
commit 728c8560c7
2 changed files with 10 additions and 8 deletions
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14
src/thesis/chapters/2_fundamentals.tex
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@@ -546,13 +546,15 @@ The \acp{vn} additionally receive messages \cite[5.4.2]{ryan_channel_2009}
computed from the channel outputs. computed from the channel outputs.
The consolidation of the information occurs in the \ac{vn} update The consolidation of the information occurs in the \ac{vn} update
\begin{align*} \begin{align*}
L_{i\rightarrow j} = \tilde{L}_i + \sum_{j'\in \mathcal{N}(i)\setminus L_{i\rightarrow j} = \tilde{L}_i + \sum_{j'\in
j} L_{i\leftarrow j'} \mathcal{N}_\text{V}(i)\setminus
\{j\}} L_{i\leftarrow j'}
\end{align*} \end{align*}
and the \ac{cn} update and the \ac{cn} update
\begin{align*} \begin{align*}
L_{i\leftarrow j} = 2\cdot \tanh^{-1} \left( \prod_{i'\in 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*} \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 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} \ac{cn} update is approximated as \cite[Sec.~5.5.1]{ryan_channel_2009}
\begin{align*} \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) \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 L_{i'\rightarrow j} \rvert
. .
\end{align*} \end{align*}
@@ -1568,7 +1570,7 @@ Additionally, we amend the \ac{cn} update to consider the parity
indicated by the syndrome, calculating indicated by the syndrome, calculating
\begin{align*} \begin{align*}
L_{i\leftarrow j} = 2\cdot (-1)^{s_j} \cdot \tanh^{-1} \left( \prod_{i'\in 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*} \end{align*}
The resulting syndrome-based \ac{bp} algorithm is shown in The resulting syndrome-based \ac{bp} algorithm is shown in

4
src/thesis/chapters/4_decoding_under_dems.tex
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@@ -899,8 +899,8 @@ To see how we realize this in practice, we reiterate the steps of the
\right) \\[3mm] \right) \\[3mm]
\text{\ac{cn} Update (Min-Sum): }& \text{\ac{cn} Update (Min-Sum): }&
\displaystyle L_{i \leftarrow j} = (-1)^{s_j}\cdot \prod_{i' \displaystyle L_{i \leftarrow j} = (-1)^{s_j}\cdot \prod_{i'
\in \mathcal{N}(j)\setminus \{i\}} \sign \left( L_{i' \rightarrow j} \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 \right) \cdot \min_{i' \in \mathcal{N}_\text{C}(j)\setminus \{i\}} \lvert
L_{i'\rightarrow j} \rvert \\[3mm] L_{i'\rightarrow j} \rvert \\[3mm]
\label{eq:vn_update} \label{eq:vn_update}
\text{\ac{vn} Update: } & \displaystyle L_{i \rightarrow j} = \text{\ac{vn} Update: } & \displaystyle L_{i \rightarrow j} =