From 455557066570b6fffc166434dad0c8d6eca75054 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Fri, 1 May 2026 18:08:57 +0200 Subject: [PATCH] Finish index definitions --- src/thesis/chapters/4_decoding_under_dems.tex | 78 +++++++++---------- 1 file changed, 39 insertions(+), 39 deletions(-) diff --git a/src/thesis/chapters/4_decoding_under_dems.tex b/src/thesis/chapters/4_decoding_under_dems.tex index 9574ae6..200cdfa 100644 --- a/src/thesis/chapters/4_decoding_under_dems.tex +++ b/src/thesis/chapters/4_decoding_under_dems.tex @@ -482,6 +482,8 @@ Finally, we call $\mathcal{N}_\text{V}(i) = \left\{ i\in \mathcal{I}: corresponding nodes. In this case, we take $\bm{H} \in \mathbb{F}_2^{m\times n}$ to be the check matrix of the underlying code, from which the \ac{dem} was generated. +We use $m_\text{DEM}, \mathcal{I}_\text{DEM}$, and $\mathcal{J}_\text{DEM}$ +to refer to the respective values defined from the detector error matrix. % How we get the corresponding rows @@ -495,23 +497,17 @@ Similarly, because of the way we defined the step size $W$, the number of \acp{cn} should be $Wm$ for all but the last window. The number of \acp{cn} in the last window may differ if there are not enough \acp{cn} left to completely fill it. -We thus define% -\footnote{ - Note that the inequality is written in terms of $j-1$, not just $j$. - This is done to transform between zero-based and one-based indexing. -} -\red{ - \begin{align*} - \mathcal{J}_\text{win}^{(\ell)} - &:= \left\{ j\in \mathcal{J}:~ \ell F m \le j - 1 < (\ell F + - W)\cdot m \right\} \\[2mm] - & \hspace{30mm} \text{and} \\[2mm] - \mathcal{J}_\text{commit}^{(\ell)} - &:= \left\{ j\in \mathcal{J}:~ \ell F m \le j - 1 < (\ell + 1)\cdot - F m \right\} - .% - \end{align*} -}% +We thus define +\begin{align*} + \mathcal{J}_\text{win}^{(\ell)} &:= \left\{ j\in \mathcal{J}_\text{DEM}:~ + \ell F m \le j < \min \left\{m_\text{DEM}, (\ell F + W) m \right\} + \right\} \\[2mm] + & \hspace{30mm} \text{and} \\[2mm] + \mathcal{J}_\text{commit}^{(\ell)} &:= \left\{ j\in \mathcal{J}_\text{DEM}:~ + \ell F m \le j < \min \left\{m_\text{DEM}, (\ell + 1) F m \right\} + \right\} + .% +\end{align*} $\mathcal{J}_\text{win}^{(\ell)}$ is the set of all \acp{cn} in the window while $\mathcal{J}_\text{commit}^{(\ell)}$ is the set of \acp{cn} that do not contribute to the next window and whose neighboring @@ -522,39 +518,44 @@ that do not contribute to the next window and whose neighboring We can now turn our attention to defining the sets of \acp{vn} relevant to each window. We first introduce a helper function $i_\text{max} : -\mathcal{P}(\mathbb{N}) \mapsto \mathbb{N}$, which takes a set of +\mathcal{P}(\mathbb{N}) \to \mathbb{N}$, which takes a set of \ac{cn} indices and returns the largest neighboring \ac{vn} index. We define \begin{align*} i_\text{max}\left( \mathcal{S} \right) := \max \left\{ i\in \mathcal{N}_\text{C}(j) : j\in \mathcal{S} \right\} - . + , \end{align*} +where we set $i_\text{max} (\emptyset) = -1$ by convention% +\footnote{ + This has the effect of later automatically setting the lower + bounds for the indices in $\mathcal{I}_\text{commit}^{(\ell)}$ + and $\mathcal{I}_\text{win}^{(\ell)}$ appropriately. +}% +. The commit region of window $\ell$ should include all of the \acp{vn} neighboring any of the \acp{cn} in $\mathcal{J}_\text{commit}^{(\ell)}$. Consequently, the maximum index of the \acp{vn} we consider should be -$i_\text{max}(\mathcal{J}_\text{commit}^{\ell})$. +$i_\text{max}(\mathcal{J}_\text{commit}^{(\ell)})$. Additionally, the set of \acp{vn} comitted in the next window should start immediately afterwards. We thus define -\red{ - \begin{align*} - \mathcal{I}_\text{commit}^{(\ell)} - &:= \left\{i \in \mathcal{I} :~ - i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right) - < i - 1 \le - i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell)} \right) - \right\}\\[2mm] - & \hspace{39mm} \text{and} \\[2mm] - \mathcal{I}_\text{win}^{(\ell)} - &:= \left\{i \in \mathcal{I} :~ - i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right) - < i - 1 \le - i_\text{max}\left( \mathcal{J}_\text{win}^{(\ell)} \right) - \right\} - .% - \end{align*} -}% +\begin{align*} + \mathcal{I}_\text{commit}^{(\ell)} + &:= \left\{i \in \mathcal{I}_\text{DEM} :~ + i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right) + < i \le + i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell)} \right) + \right\}\\[2mm] + & \hspace{39mm} \text{and} \\[2mm] + \mathcal{I}_\text{win}^{(\ell)} + &:= \left\{i \in \mathcal{I}_\text{DEM} :~ + i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right) + < i \le + i_\text{max}\left( \mathcal{J}_\text{win}^{(\ell)} \right) + \right\} + .% +\end{align*} Note that we have \begin{align*} \bigcup_{\ell=0}^{n_\text{win}-1} @@ -564,7 +565,6 @@ and after decoding all windows we will therefore have committed all \acp{vn}. % Syndrome update -\content{Explain commit region} \content{Why we need to update the syndrome} \content{How we update the syndrome} \content{\textbf{General note}: Mathematical definitions where possible}