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