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Start VN and CN indexing from zero

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This commit is contained in:
Andreas Tsouchlos
2026-05-01 17:30:14 +02:00
parent 635c0aab18
commit 3b7618e1d1
3 changed files with 20 additions and 22 deletions
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26
src/thesis/chapters/2_fundamentals.tex
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@@ -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}