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Co-authored-by: Mark <mark@betalupi.com>
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\section{Error Correction}
As we saw in \ref{isbn-nocorrect}, the ISBN check-digit scheme does not allow us to correct errors. \par
QR codes feature a system that does. \par
\vspace{1mm}
Odds are, you've seen a QR code with an image in the center. Such codes aren't \say{special}---they're simply missing their central pixels. The error-correcting algorithm in the QR specification allows us to read the code despite this damage.
\begin{figure}[h]
\centering
\href{https://youtube.com/watch?v=dQw4w9WgXcQ}{\includegraphics[width = 3cm]{qr}}
\end{figure}
\definition{Repeating codes}
The simplest possible error-correcting code is a \textit{repeating code}. It works just as you'd expect: \par
Instead of sending data once, it sends multiple copies of each bit. \par
If a few bits are damaged, they can be both detected and repaired. \par
For example, consider the following three-repeat code encoding the binary string $101$:
$$
111~000~111
$$
If we flip any one bit, we can easily find and fix the error.
\problem{}<number-repeat>
How many repeated digits do you need to...
\begin{itemize}
\item[-] detect a transposition error?
\item[-] correct a transposition error?
\end{itemize}
\vfill
\definition{Code Efficiency}
The efficiency of an error-correcting code is calculated as follows:
$$
\frac{\text{number of data bits}}{\text{total bits sent}}
$$
For example, the efficiency of the three-repeat code above is $\frac{3}{9} = \frac{1}{3} \approx 0.33$
\problem{}<k-efficiency>
What is the efficiency of a $k$-repeat code?
\vfill
As you just saw, repeat codes are not a good solution. You need many extra bits for even a small amount of redundancy. We need a better system.
\pagebreak
%\definition{Hamming's Square Code}
%We will now analyze a more efficient coding scheme: \par
%
%\vspace{1mm}
%
%Take a four-bit message and arrange it in a $2 \times 2$ square. \par
%Compute the pairity of each row and write it at the right. \par
%Compute the pairity of each column and write it at the bottom. \par
%Finally, compute the pairity of the entire message write it in the lower right corner.
%This ensures that the total number of ones in the message is even.
%
%\vspace{2mm}
%
%Reading the result row by row to get the encoded message. \par
%For example, the message 1011 generates the sequence 101110011:
%
%$$
%1011
%\longrightarrow
%\begin{array}{cc|}
% 1 & 0 \\
% 1 & 1 \\
% \hline
%\end{array}
%\longrightarrow
%\begin{array}{cc|c}
% 1 & 0 & 1 \\
% 1 & 1 & 0 \\ \hline
% 0 & 1 &
%\end{array}
%\longrightarrow
%\begin{array}{cc|c}
% 1 & 0 & 1 \\
% 1 & 1 & 0 \\ \hline
% 0 & 1 & 1
%\end{array}
%\longrightarrow
%101110011
%$$
%
%\problem{}
%The following messages are encoded using the method above.
%Find and correct any single-digit or transposition errors.
%\begin{enumerate}
% \item \texttt{110 110 011} %101110011
% \item \texttt{100 101 011} %110101011
% \item \texttt{001 010 110} %000110110
%\end{enumerate}
%
%\begin{solution}
% \begin{enumerate}
% \item \texttt{101 110 011} or \texttt{110 101 011}
% \item \texttt{110 101 011}
% \item \texttt{000 110 110}
% \end{enumerate}
%\end{solution}
%
%\vfill
%
%\problem{}
%What is the efficiency of this coding scheme?
%
%\vfill
%
%\problem{}
%Can we correct a single-digit error in the encoded message? \par
%Can we correct a transposition error in the encoded message?
%
%\vfill
%
%\problem{}
%Let's generalize this coding scheme to a non-square table: \par
%Given a message of length $ab$, construct a rectangle with dimensions $a \times b$ as described above.
%\begin{itemize}
% \item What is the efficiency of a $a \times b$ rectangle code?
% \item Can the $a \times b$ rectangle code detect and fix single-bit errors?
% \item Can the $a \times b$ rectangle code detect and fix two-bit errors?
%\end{itemize}
%
%\vfill
%\pagebreak