Mark/handouts
1
0
Fork 0
Code Pull Requests 1 Actions Packages Releases Activity

Improved ECC handout

Browse Source
This commit is contained in:
Mark
2023-06-19 15:42:33 -07:00
parent e2c1e72e3a
commit 20c0a84a0c
6 changed files with 591 additions and 185 deletions
Download Patch File Download Diff File Expand all files Collapse all files

53
Advanced/Error-Correcting Codes/parts/01 correction.tex
View File

@ -1,14 +1,20 @@
\section{Error Correction}
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.
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 \say{repeating code}. It works just as you'd expect: \\
Instead of sending data once, it sends multiple copies. If a few bits are damaged, they can be both detected and repaired. \\
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 $101$:
@ -18,6 +24,16 @@ $$
If we flip any one bit, we can easily find and fix the error.
\problem{}
How many repeated digits do you need to...
\begin{itemize}
\item[-] detect a transposition error?
\item[-] correct a transposition error?
\end{itemize}
\vfill
\definition{Code Efficiency}
The efficiency of an error-correcting code is calculated as follows:
$$
@ -27,32 +43,25 @@ $$
For example, the efficiency of the three-repeat code above is $\frac{3}{9} = \frac{1}{3} \approx 0.33$
\problem{}
What is the efficiency of an $k$-repeat code?
\vfill
\problem{}
How many repeated digits do you need to...
\begin{itemize}
\item[-] detect a transposition error?
\item[-] correct a transposition error?
\end{itemize}
What is the efficiency of a $k$-repeat code?
\vfill
\pagebreak
\definition{Hamming's Square Code}
We will now analyze a more efficient coding scheme: \par
A more effective coding scheme comes in the form of Hamming's square code.
Take a four-bit message and arrange it in a $2 \times 2$ square. \\
\vspace{1mm}
Compute the pairity of each row and write it at the right. \\
Compute the pairity of each column and write it at the bottom. \\
Take a four-bit message and arrange it in a $2 \times 2$ square. \par
Compute the pairity of each row and write it at the right. \par
Compute the pairity of each column and write it at the bottom. \par
Finally, compute the pairity of the entire message write it in the lower right corner.
This ensures that the total number of ones in the message is even.
\vspace{3mm}
\vspace{2mm}
Reading the result row by row to get the encoded message. \\
Reading the result row by row to get the encoded message. \par
For example, the message 1011 generates the sequence 101110011:
$$
@ -80,7 +89,7 @@ $$
$$
\problem{}
The following message are encoded using the method above.
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
@ -104,13 +113,13 @@ What is the efficiency of this coding scheme?
\vfill
\problem{}
Can we correct a single-digit error in the encoded message? \\
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: \\
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?