Added ECC handout

Advanced/Error-Correcting Codes/parts/02 distance.tex

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+\section{Hamming Distance}
+
+\definition{}
+The \textit{Hamming distance} between two strings $x = x_1x_2...x_n$ and $y = y_1y_2...y_n$ is the number of positions at which the digits of $x$ and $y$ are different.
+
+\problem{}
+Compute the Hamming distance between \texttt{1010} and \texttt{0001}.
+
+\vfill
+
+\problem{}
+Read $d_H(x, y)$ as \say{the hamming distance between $x$ and $y$.} \\
+Prove the following statements:
+\begin{enumerate}
+\item $d_H(x, y) \ge 0$ with equality if and only if $x = y$,
+\item $d_H(x, y) = d_H(y, x)$,
+\item $d_H(x, z) \le d_H(x, y) + d_H(y, z)$.
+\end{enumerate}
+
+\vfill
+
+\problem{}
+Say we encode and send a message with the 3-repeat code. A few bits are damaged in transit. \\
+When the transmission is decoded, a different message is read.
+
+\vspace{2mm}
+
+What is the minimum possible hamming distance between the undamaged encoded message and the damaged encoded message?
+
+\vfill
+
+\problem{}
+Say we encode and send a message with Hamming's square code. A few bits are damaged in transit. \\
+When the transmission is decoded, no uncorrectable errors are detected and a different message is read.
+
+\vspace{2mm}
+
+What is the minimum possible hamming distance between the undamaged encoded message and the damaged encoded message?
+
+\vfill
+\pagebreak