123 lines
3.3 KiB
TeX
123 lines
3.3 KiB
TeX
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\section{Error Correction}
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Error detection is helpful, but we'd also like to fix errors when we find them. One example of such a system is the QR code, which remains readable even if a significant amount of it is removed. QR codes with icons inside aren't special--they're just missing their central elements. The error-correcting codes in the QR specification allow us to recover the lost data.
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\begin{figure}[h]
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\centering
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\includegraphics[width = 3cm]{qr}
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\end{figure}
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\definition{Repeating codes}
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The simplest possible error-correcting code is a \say{repeating code}. It works just as you'd expect: \\
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Instead of sending data once, it sends multiple copies. If a few bits are damaged, they can be both detected and repaired. \\
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For example, consider the following three-repeat code encoding the binary string $101$:
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$$
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111~000~111
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$$
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If we flip any one bit, we can easily find and fix the error.
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\definition{Code Efficiency}
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The efficiency of an error-correcting code is calculated as follows:
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$$
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\frac{\text{number of data bits}}{\text{total bits sent}}
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$$
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For example, the efficiency of the three-repeat code above is $\frac{3}{9} = \frac{1}{3} \approx 0.33$
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\problem{}
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What is the efficiency of an $k$-repeat code?
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\vfill
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\problem{}
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How many repeated digits do you need to...
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\begin{itemize}
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\item[-] detect a transposition error?
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\item[-] correct a transposition error?
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\end{itemize}
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\vfill
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\pagebreak
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\definition{Hamming's Square Code}
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A more effective coding scheme comes in the form of Hamming's square code.
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Take a four-bit message and arrange it in a $2 \times 2$ square. \\
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Compute the pairity of each row and write it at the right. \\
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Compute the pairity of each column and write it at the bottom. \\
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Finally, compute the pairity of the entire message write it in the lower right corner.
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\vspace{3mm}
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Reading the result row by row to get the encoded message. \\
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For example, the message 1011 generates the sequence 101110011:
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$$
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1011
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\longrightarrow
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\begin{array}{cc|}
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1 & 0 \\
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1 & 1 \\
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\hline
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\end{array}
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\longrightarrow
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\begin{array}{cc|c}
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1 & 0 & 1 \\
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1 & 1 & 0 \\ \hline
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0 & 1 &
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\end{array}
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\longrightarrow
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\begin{array}{cc|c}
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1 & 0 & 1 \\
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1 & 1 & 0 \\ \hline
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0 & 1 & 1
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\end{array}
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\longrightarrow
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101110011
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$$
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\problem{}
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The following message are encoded using the method above.
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Find and correct any single-digit or transposition errors.
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\begin{enumerate}
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\item \texttt{110 110 011} %101110011
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\item \texttt{100 101 011} %110101011
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\item \texttt{001 010 110} %000110110
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\end{enumerate}
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\begin{solution}
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\begin{enumerate}
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\item \texttt{101 110 011} or \texttt{110 101 011}
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\item \texttt{110 101 011}
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\item \texttt{000 110 110}
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\end{enumerate}
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\end{solution}
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\vfill
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\problem{}
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What is the efficiency of this coding scheme?
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\vfill
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\problem{}
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Can we correct a single-digit error in the encoded message? \\
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Can we correct a transposition error in the encoded message?
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\vfill
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\problem{}
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Let's generalize this coding scheme to a non-square table: \\
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Given a message of length $ab$, construct a rectangle with dimensions $a \times b$ as described above.
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\begin{itemize}
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\item What is the efficiency of a $a \times b$ rectangle code?
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\item Can the $a \times b$ rectangle code detect and fix single-bit errors?
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\item Can the $a \times b$ rectangle code detect and fix two-bit errors?
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\end{itemize}
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\vfill
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\pagebreak
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