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Final readthrough corrections of classical fundamentals

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Andreas Tsouchlos
2026-05-04 21:07:25 +02:00
parent 400dc47df0
commit 06852b8e62
1 changed files with 3 additions and 3 deletions
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src/thesis/chapters/2_fundamentals.tex
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@@ -39,7 +39,7 @@ Binary linear block codes form one particularly important class of
coding schemes. coding schemes.
The information to be protected is represented by a sequence of The information to be protected is represented by a sequence of
binary symbols, which is split into separate blocks. binary symbols, which is split into separate blocks.
Then, each block is encoded, transmitted, and decoded separately. Each block is encoded, transmitted, and decoded separately.
The encoding step introduces redundancy by mapping input messages The encoding step introduces redundancy by mapping input messages
$\bm{u} \in \mathbb{F}_2^k$ of length $k \in \mathbb{N}$ (called the $\bm{u} \in \mathbb{F}_2^k$ of length $k \in \mathbb{N}$ (called the
\textit{information length}) onto \textit{codewords} $\bm{x} \in \textit{information length}) onto \textit{codewords} $\bm{x} \in
@@ -276,10 +276,10 @@ and a \ac{cn} using the index $j \in \mathcal{J}
:= \left[ 0 : m-1 \right]$. := \left[ 0 : m-1 \right]$.
We can then encode the information contained in the graph by defining We can then encode the information contained in the graph by defining
the neighborhood of a \ac{vn} $i$ as the neighborhood of a \ac{vn} $i$ as
$\mathcal{N}_\text{V} (i) = \left\{ j \in \mathcal{J} : \bm{H}_{j,i} $\mathcal{N}_\text{V} (i) = \left\{ j \in \mathcal{J} : H_{j,i}
= 1 \right\}$ = 1 \right\}$
and the neighborhood of a \ac{cn} $j$ as and the neighborhood of a \ac{cn} $j$ as
$\mathcal{N}_\text{C} (j) = \left\{ i \in \mathcal{I} : \bm{H}_{j,i} $\mathcal{N}_\text{C} (j) = \left\{ i \in \mathcal{I} : H_{j,i}
= 1 \right\}$. = 1 \right\}$.
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