\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
	\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{}
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{}
What is the efficiency of a $k$-repeat code?

\vfill
\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
