update

correction.tex

@@ -30,6 +30,8 @@
 \maketitle
 \section{Introduction}
 When messages are transmitted over real media, errors are inevitably introduced.
+Examples range from mundane sources of errors like scratches on CDs and capacitive coupling of conductors
+to geomagnetic storms.
 Though the source of errors are manifold, they are ubiquitous and need to be accounted for if we want any
 real chance of reading a correct message at the end.
 On a basic level, we can understand that a channel with a lower \textit{\acrfull{snr}},
@@ -49,7 +51,60 @@ If the noise level is small on average, most transmissions will be interpreted c
 However, there generally remains a chance that an intended 1 will be received as .47 volts and hence interpreted incorrectly as a 0.
 \cite{enwiki:signal-to-noise}
 
+
 In this digital system, we can improve communication reliability by using a coding scheme that is tolerant of errors.
+As a naive approach, we will simply transmit our message multiple times.
+This is called a \textit{repetition code}, where the receiver will perform a majority vote over the received bits,
+resulting in the following
+
+\begin{figure}[H]
+\begin{minipage}{.5\textwidth}
+    \begin{tabular}{c|ccc}
+        recieved & 0 & \textcolor{red}{1} & 2 \\
+        \hline
+        read & 0 & ERR &  1 \\
+    \end{tabular}
+
+    \begin{tabular}{c|cccccccc}
+        recieved & 0 & \textcolor{red}{1} & \textcolor{red}{2} & 3 \\
+        \hline
+        read & 0 & 0 & 1 & 1  \\
+    \end{tabular}
+
+    \begin{tabular}{c|cccccccc}
+        recieved & 0 & \textcolor{red}{1} & \textcolor{red}{2} & \textcolor{red}{3} & 4 \\
+        \hline
+        read & 0 & 0 & ERR & 1 & 1 \\
+    \end{tabular}
+\caption{Error detection and detection for 2,3 and 4 bits used per symbol in a repetition code}
+\label{tab:detection-correction}
+\end{minipage}
+\begin{minipage}{.5\textwidth}
+\begin{tikzpicture}
+  \begin{axis}[
+    domain=0:1,
+    samples=100,
+    axis lines=middle,
+    xlabel={$p$},
+    ylabel={Channel Capacity $C$},
+    xmin=0, xmax=1,
+    ymin=0, ymax=1.1,
+    xtick={0,0.25,0.5,0.75,1},
+    grid=both,
+    width=8cm,
+    height=6cm,
+    every axis x label/.style={at={(current axis.right of origin)}, anchor=west},
+    every axis y label/.style={at={(current axis.above origin)}, anchor=south},
+  ]
+    \addplot[thick, blue] {1-(-x * log2(x) - (1-x) * log2(1-x))};
+  \end{axis}
+\end{tikzpicture}
+\caption{Capacity of a binary channel with crossover probability $p$}
+\label{fig:graph-errorrate}
+\end{minipage}
+\end{figure}
+
+
 
 \section{Hamming Condition}
 theorems: Hamming condition, Varsham-Gilbert