\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