Finish index definitions

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%
+We thus define
-\footnote{
+\begin{align*}
-    Note that the inequality is written in terms of $j-1$, not just $j$.
+    \mathcal{J}_\text{win}^{(\ell)} &:= \left\{ j\in \mathcal{J}_\text{DEM}:~
-    This is done to transform between zero-based and one-based indexing.
+        \ell F m \le j < \min \left\{m_\text{DEM}, (\ell F + W) m \right\}
-}
+    \right\} \\[2mm]
-\red{
+    & \hspace{30mm} \text{and} \\[2mm]
-    \begin{align*}
+    \mathcal{J}_\text{commit}^{(\ell)} &:= \left\{ j\in \mathcal{J}_\text{DEM}:~
-        \mathcal{J}_\text{win}^{(\ell)}
+        \ell F m \le j < \min \left\{m_\text{DEM}, (\ell + 1) F m \right\}
-        &:= \left\{ j\in \mathcal{J}:~ \ell F m \le j - 1 < (\ell F +
+    \right\}
-        W)\cdot m \right\} \\[2mm]
+    .%
-        & \hspace{30mm} \text{and} \\[2mm]
+\end{align*}
-        \mathcal{J}_\text{commit}^{(\ell)}
-        &:= \left\{ j\in \mathcal{J}:~ \ell F m \le j - 1 < (\ell + 1)\cdot
-        F m \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*}
-    \begin{align*}
+    \mathcal{I}_\text{commit}^{(\ell)}
-        \mathcal{I}_\text{commit}^{(\ell)}
+    &:= \left\{i \in \mathcal{I}_\text{DEM} :~
-        &:= \left\{i \in \mathcal{I} :~
+        i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
-            i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
+        < i \le
-            < i - 1 \le
+        i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell)} \right)
-            i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell)} \right)
+    \right\}\\[2mm]
-        \right\}\\[2mm]
+    & \hspace{39mm} \text{and} \\[2mm]
-        & \hspace{39mm} \text{and} \\[2mm]
+    \mathcal{I}_\text{win}^{(\ell)}
-        \mathcal{I}_\text{win}^{(\ell)}
+    &:= \left\{i \in \mathcal{I}_\text{DEM} :~
-        &:= \left\{i \in \mathcal{I} :~
+        i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
-            i_\text{max}\left( \mathcal{J}_\text{commit}^{(\ell-1)} \right)
+        < i \le
-            < i - 1 \le
+        i_\text{max}\left( \mathcal{J}_\text{win}^{(\ell)} \right)
-            i_\text{max}\left( \mathcal{J}_\text{win}^{(\ell)} \right)
+    \right\}
-        \right\}
+    .%
-        .%
+\end{align*}
-    \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}