diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex index a9acd07..42bb2ad 100644 --- a/src/thesis/chapters/2_fundamentals.tex +++ b/src/thesis/chapters/2_fundamentals.tex @@ -39,7 +39,7 @@ Binary linear block codes form one particularly important class of coding schemes. The information to be protected is represented by a sequence of 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 $\bm{u} \in \mathbb{F}_2^k$ of length $k \in \mathbb{N}$ (called the \textit{information length}) onto \textit{codewords} $\bm{x} \in @@ -276,10 +276,10 @@ and a \ac{cn} using the index $j \in \mathcal{J} := \left[ 0 : m-1 \right]$. We can then encode the information contained in the graph by defining 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\}$ 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\}$. %