Fixed errors

Advanced/Error-Correcting Codes/parts/00 detection.tex

@@ -5,26 +5,25 @@ An ISBN\footnote{International Standard Book Number} is a unique numeric book id
 \vspace{3mm}
 
 Say we have a sequence of nine digits, forming a partial ISBN-10: $n_1 n_2 ... n_9$. \par
-The final digit, $n_{10}$, is calculated as follows:
+The final digit, $n_{10}$, is chosen from $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ so that:
 
 $$
-	\Biggr( \sum_{i = 1}^{9} (11 - i) \times n_i \Biggl) \text{ mod } 11
+	\sum_{i = 1}^{10} (11 - i)n_i
 $$
 
 If $n_{10}$ is equal to 10, it is written as \texttt{X}.
 
 
 \problem{}
-Which of the following could be valid ISBNs?
+Only one of the following ISBNs is valid. Which one is it?
 
 \begin{itemize}
 	\item \texttt{0-134-54896-2}
 	\item \texttt{0-895-77258-2}
-	\item \texttt{0-316-00395-6}
 \end{itemize}
 
 \begin{solution}
-	Only the first has an inconsistent check digit.
+	The first has an inconsistent check digit.
 \end{solution}
 
 \vfill
@@ -66,31 +65,6 @@ This is called a \textit{transposition error}.
 \end{solution}
 
 \vfill
-
-
-
-\problem{}
-Show that the following sum is divisible by 11 iff $n_1n_2...n_{10}$ is a valid ISBN-10.
-$$
-	\sum_{i = 1}^{10} (11 - i)n_i
-$$
-
-\begin{solution}
-	Proof that valid $\implies$ divisible, working in mod 11:
-
-	\vspace{2mm}
-
-	$10n_1 + 9n_2 + ... + 2n_9 + n_{10} \equiv$ \par
-	$(-n_1) + (-2n_2) + ... + (-9n_9) + n_{10} =$ \par
-	$-n_{10} + n_{10} \equiv 0$
-
-	\vspace{2mm}
-
-	Having done this, the rest is easy. Work in reverse, or note that each step above is an iff.
-
-\end{solution}
-
-\vfill
 \pagebreak
 
 \problem{}

Advanced/Error-Correcting Codes/parts/02 hamming.tex

@@ -76,14 +76,54 @@ How many message bits have an index with a one as the $n^\text{th}$ digit? \par
 
 
 
-We now want a way to detect errors in our 16-bit message. To do this, we'll replace a few data bits with pairity bits. This will reduce the amount of information we can send, but will also improve our error-detection capabilities.
+We now want a way to detect errors in our 16-bit message. To do this, we'll replace a few data bits with parity bits. This will reduce the amount of information we can send, but will also improve our error-detection capabilities.
 
 \vspace{1mm}
 
-Let's arrange our message in a grid. We'll make the first bit (currently empty, marked \texttt{X}) a pairity bit. Its value will depend on the content of the message: if our message has an even number of ones, it will be zero; if our message has an odd number of ones, it will be one.
+Let's arrange our message in a grid. We'll make the first bit (currently empty, marked \texttt{X}) a parity bit. Its value will depend on the content of the message: if our message has an even number of ones, it will be zero; if our message has an odd number of ones, it will be one. \par
+This first bit ensures that there is an even number of ones in the whole message.
 
 \begin{center}
+\hfill
 \begin{tikzpicture}[scale = 1.25]
+	\node at (0.75, 0.5) {Bit Numbering};
+
+	\node at (0.0, 0) {\texttt{0}};
+	\node at (0.5, 0) {\texttt{1}};
+	\node at (1.0, 0) {\texttt{2}};
+	\node at (1.5, 0) {\texttt{3}};
+
+	\node at (0.0, -0.5) {\texttt{4}};
+	\node at (0.5, -0.5) {\texttt{5}};
+	\node at (1.0, -0.5) {\texttt{6}};
+	\node at (1.5, -0.5) {\texttt{7}};
+
+	\node at (0.0, -1) {\texttt{8}};
+	\node at (0.5, -1) {\texttt{9}};
+	\node at (1.0, -1) {\texttt{10}};
+	\node at (1.5, -1) {\texttt{11}};
+
+	\node at (0.0, -1.5) {\texttt{12}};
+	\node at (0.5, -1.5) {\texttt{13}};
+	\node at (1.0, -1.5) {\texttt{14}};
+	\node at (1.5, -1.5) {\texttt{15}};
+
+	\draw (-0.25, 0.25) -- (1.75, 0.25);
+	\draw (-0.25, -0.25) -- (1.75, -0.25);
+	\draw (-0.25, -0.75) -- (1.75, -0.75);
+	\draw (-0.25, -1.25) -- (1.75, -1.25);
+	\draw (-0.25, -1.75) -- (1.75, -1.75);
+
+	\draw (-0.25, 0.25) -- (-0.25, -1.75);
+	\draw (0.25, 0.25) -- (0.25, -1.75);
+	\draw (0.75, 0.25) -- (0.75, -1.75);
+	\draw (1.25, 0.25) -- (1.25, -1.75);
+	\draw (1.75, 0.25) -- (1.75, -1.75);
+\end{tikzpicture}
+\hfill
+\begin{tikzpicture}[scale = 1.25]
+	\node at (0.75, 0.5) {Sample Message};
+
 	\node at (0.0, 0) {\texttt{X}};
 	\node at (0.5, 0) {\texttt{0}};
 	\node at (1.0, 0) {\texttt{1}};
@@ -118,10 +158,11 @@ Let's arrange our message in a grid. We'll make the first bit (currently empty,
 
 	\draw (-0.2,-0.2) -- (0.2, -0.2) -- (0.2, 0.2) -- (-0.2, 0.2) -- (-0.2,-0.2);
 \end{tikzpicture}
+\hfill~
 \end{center}
 
 \problem{}
-What is the value of the pairity bit in the message above?
+What is the value of the parity bit in the message above?
 
 \vfill
 
@@ -133,7 +174,7 @@ Can this coding scheme correct a single-bit error?
 \vfill
 \pagebreak
 
-We'll now add four more pairity bits, in positions \texttt{0001}, \texttt{0010}, \texttt{0100}, and \texttt{1000}:
+We'll now add four more parity bits, in positions \texttt{0001}, \texttt{0010}, \texttt{0100}, and \texttt{1000}:
 
 \begin{center}
 \begin{tikzpicture}[scale = 1.25]
@@ -198,18 +239,128 @@ We'll now add four more pairity bits, in positions \texttt{0001}, \texttt{0010},
 \end{center}
 
 
-Bit \texttt{0001} will count the pairity of all bits with a one in the first digit of their index. \par
+Bit \texttt{0001} will count the parity of all bits with a one in the first digit of their index. \par
-Bit \texttt{0010} will count the pairity of all bits with a one in the second digit of their index. \par
+Bit \texttt{0010} will count the parity of all bits with a one in the second digit of their index. \par
-Bits \texttt{0100} and \texttt{1000} work in the same way.
+Bits \texttt{0100} and \texttt{1000} work in the same way. \par
+\hint{In \texttt{0001}, \texttt{1} is the first digit. In \texttt{0010}, \texttt{1} is the second digit. \\
+When counting bits in binary numbers, go from right to left.}
 
 \problem{}
-Compute all pairity bits in the message above.
+Which message bits does each parity bit cover? \par
+In other words, which message bits affect the value of each parity bit? \par
+
+\vspace{1mm}
+
+Four diagrams are shown below. In each grid, fill in the bits that affect the shaded parity bit.
+
+\begin{center}
+	\hfill
+	\begin{tikzpicture}[scale = 1.25]
+		\draw (-0.25, 0.25) -- (1.75, 0.25);
+		\draw (-0.25, -0.25) -- (1.75, -0.25);
+		\draw (-0.25, -0.75) -- (1.75, -0.75);
+		\draw (-0.25, -1.25) -- (1.75, -1.25);
+		\draw (-0.25, -1.75) -- (1.75, -1.75);
+
+		\draw (-0.25, 0.25) -- (-0.25, -1.75);
+		\draw (0.25, 0.25) -- (0.25, -1.75);
+		\draw (0.75, 0.25) -- (0.75, -1.75);
+		\draw (1.25, 0.25) -- (1.25, -1.75);
+		\draw (1.75, 0.25) -- (1.75, -1.75);
+
+		\draw[pattern=north east lines] (0.5 - 0.2, 0 - 0.2)
+			-- (0.5 + 0.2, 0 - 0.2)
+			-- (0.5 + 0.2, 0 + 0.2)
+			-- (0.5 - 0.2, 0 + 0.2)
+			-- (0.5 - 0.2, 0 - 0.2);
+	\end{tikzpicture}
+	\hfill
+	\begin{tikzpicture}[scale = 1.25]
+		\draw (-0.25, 0.25) -- (1.75, 0.25);
+		\draw (-0.25, -0.25) -- (1.75, -0.25);
+		\draw (-0.25, -0.75) -- (1.75, -0.75);
+		\draw (-0.25, -1.25) -- (1.75, -1.25);
+		\draw (-0.25, -1.75) -- (1.75, -1.75);
+
+		\draw (-0.25, 0.25) -- (-0.25, -1.75);
+		\draw (0.25, 0.25) -- (0.25, -1.75);
+		\draw (0.75, 0.25) -- (0.75, -1.75);
+		\draw (1.25, 0.25) -- (1.25, -1.75);
+		\draw (1.75, 0.25) -- (1.75, -1.75);
+
+
+		\draw[pattern=north east lines] (1 - 0.2, 0 - 0.2)
+			-- (1 + 0.2, 0 - 0.2)
+			-- (1 + 0.2, 0 + 0.2)
+			-- (1 - 0.2, 0 + 0.2)
+			-- (1 - 0.2, 0 - 0.2);
+	\end{tikzpicture}
+	\hfill
+	\begin{tikzpicture}[scale = 1.25]
+		\draw (-0.25, 0.25) -- (1.75, 0.25);
+		\draw (-0.25, -0.25) -- (1.75, -0.25);
+		\draw (-0.25, -0.75) -- (1.75, -0.75);
+		\draw (-0.25, -1.25) -- (1.75, -1.25);
+		\draw (-0.25, -1.75) -- (1.75, -1.75);
+
+		\draw (-0.25, 0.25) -- (-0.25, -1.75);
+		\draw (0.25, 0.25) -- (0.25, -1.75);
+		\draw (0.75, 0.25) -- (0.75, -1.75);
+		\draw (1.25, 0.25) -- (1.25, -1.75);
+		\draw (1.75, 0.25) -- (1.75, -1.75);
+
+		\draw[pattern=north east lines] (0 - 0.2, -0.5 - 0.2)
+			-- (0 + 0.2, -0.5 - 0.2)
+			-- (0 + 0.2, -0.5 + 0.2)
+			-- (0 - 0.2, -0.5 + 0.2)
+			-- (0 - 0.2, -0.5 - 0.2);
+	\end{tikzpicture}
+	\hfill
+	\begin{tikzpicture}[scale = 1.25]
+		\draw (-0.25, 0.25) -- (1.75, 0.25);
+		\draw (-0.25, -0.25) -- (1.75, -0.25);
+		\draw (-0.25, -0.75) -- (1.75, -0.75);
+		\draw (-0.25, -1.25) -- (1.75, -1.25);
+		\draw (-0.25, -1.75) -- (1.75, -1.75);
+
+		\draw (-0.25, 0.25) -- (-0.25, -1.75);
+		\draw (0.25, 0.25) -- (0.25, -1.75);
+		\draw (0.75, 0.25) -- (0.75, -1.75);
+		\draw (1.25, 0.25) -- (1.25, -1.75);
+		\draw (1.75, 0.25) -- (1.75, -1.75);
+
+		\draw[pattern=north east lines] (0 - 0.2, -1 - 0.2)
+			-- (0 + 0.2, -1 - 0.2)
+			-- (0 + 0.2, -1 + 0.2)
+			-- (0 - 0.2, -1 + 0.2)
+			-- (0 - 0.2, -1 - 0.2);
+	\end{tikzpicture}
+	\hfill
+\end{center}
+
+
+
+\problem{}
+Compute all parity bits in the message above.
+
+\vfill
+\pagebreak
+
+
+\problem{}
+Analyze this coding scheme.
+\begin{itemize}
+	\item Can we detect one single-bit errors?
+	\item Can we detect two single-bit errors?
+	\item What errors can we correct?
+\end{itemize}
 
 \vfill
 
 \problem{}
 Each of the following messages has either 0, 1, or two errors. \par
-Find the errors and correct them if possible.
+Find the errors and correct them if possible. \par
+\hint{Bit \texttt{0000} should tell you how many errors you have.}
 
 \begin{center}
 \hfill
@@ -405,112 +556,9 @@ Find the errors and correct them if possible.
 \vfill
 \pagebreak
 
-\problem{}
-Analyze this coding scheme.
-\begin{itemize}
-	\item Can we detect one single-bit errors?
-	\item Can we detect two single-bit errors?
-	\item What errors can we correct?
-\end{itemize}
-
-\vfill
 
 \problem{}
-Which message bits does each pairity bit cover? \par
+How many parity bits does each message bit affect? \par
-In other words, which message bits affect the value of each pairity bit? \par
-
-\vspace{1mm}
-
-Four diagrams are shown below. In each grid, fill in the bits that affect the shaded pairity bit.
-
-\begin{center}
-	\hfill
-	\begin{tikzpicture}[scale = 1.25]
-		\draw (-0.25, 0.25) -- (1.75, 0.25);
-		\draw (-0.25, -0.25) -- (1.75, -0.25);
-		\draw (-0.25, -0.75) -- (1.75, -0.75);
-		\draw (-0.25, -1.25) -- (1.75, -1.25);
-		\draw (-0.25, -1.75) -- (1.75, -1.75);
-
-		\draw (-0.25, 0.25) -- (-0.25, -1.75);
-		\draw (0.25, 0.25) -- (0.25, -1.75);
-		\draw (0.75, 0.25) -- (0.75, -1.75);
-		\draw (1.25, 0.25) -- (1.25, -1.75);
-		\draw (1.75, 0.25) -- (1.75, -1.75);
-
-		\draw[pattern=north east lines] (0.5 - 0.2, 0 - 0.2)
-			-- (0.5 + 0.2, 0 - 0.2)
-			-- (0.5 + 0.2, 0 + 0.2)
-			-- (0.5 - 0.2, 0 + 0.2)
-			-- (0.5 - 0.2, 0 - 0.2);
-	\end{tikzpicture}
-	\hfill
-	\begin{tikzpicture}[scale = 1.25]
-		\draw (-0.25, 0.25) -- (1.75, 0.25);
-		\draw (-0.25, -0.25) -- (1.75, -0.25);
-		\draw (-0.25, -0.75) -- (1.75, -0.75);
-		\draw (-0.25, -1.25) -- (1.75, -1.25);
-		\draw (-0.25, -1.75) -- (1.75, -1.75);
-
-		\draw (-0.25, 0.25) -- (-0.25, -1.75);
-		\draw (0.25, 0.25) -- (0.25, -1.75);
-		\draw (0.75, 0.25) -- (0.75, -1.75);
-		\draw (1.25, 0.25) -- (1.25, -1.75);
-		\draw (1.75, 0.25) -- (1.75, -1.75);
-
-
-		\draw[pattern=north east lines] (1 - 0.2, 0 - 0.2)
-			-- (1 + 0.2, 0 - 0.2)
-			-- (1 + 0.2, 0 + 0.2)
-			-- (1 - 0.2, 0 + 0.2)
-			-- (1 - 0.2, 0 - 0.2);
-	\end{tikzpicture}
-	\hfill
-	\begin{tikzpicture}[scale = 1.25]
-		\draw (-0.25, 0.25) -- (1.75, 0.25);
-		\draw (-0.25, -0.25) -- (1.75, -0.25);
-		\draw (-0.25, -0.75) -- (1.75, -0.75);
-		\draw (-0.25, -1.25) -- (1.75, -1.25);
-		\draw (-0.25, -1.75) -- (1.75, -1.75);
-
-		\draw (-0.25, 0.25) -- (-0.25, -1.75);
-		\draw (0.25, 0.25) -- (0.25, -1.75);
-		\draw (0.75, 0.25) -- (0.75, -1.75);
-		\draw (1.25, 0.25) -- (1.25, -1.75);
-		\draw (1.75, 0.25) -- (1.75, -1.75);
-
-		\draw[pattern=north east lines] (0 - 0.2, -0.5 - 0.2)
-			-- (0 + 0.2, -0.5 - 0.2)
-			-- (0 + 0.2, -0.5 + 0.2)
-			-- (0 - 0.2, -0.5 + 0.2)
-			-- (0 - 0.2, -0.5 - 0.2);
-	\end{tikzpicture}
-	\hfill
-	\begin{tikzpicture}[scale = 1.25]
-		\draw (-0.25, 0.25) -- (1.75, 0.25);
-		\draw (-0.25, -0.25) -- (1.75, -0.25);
-		\draw (-0.25, -0.75) -- (1.75, -0.75);
-		\draw (-0.25, -1.25) -- (1.75, -1.25);
-		\draw (-0.25, -1.75) -- (1.75, -1.75);
-
-		\draw (-0.25, 0.25) -- (-0.25, -1.75);
-		\draw (0.25, 0.25) -- (0.25, -1.75);
-		\draw (0.75, 0.25) -- (0.75, -1.75);
-		\draw (1.25, 0.25) -- (1.25, -1.75);
-		\draw (1.75, 0.25) -- (1.75, -1.75);
-
-		\draw[pattern=north east lines] (0 - 0.2, -1 - 0.2)
-			-- (0 + 0.2, -1 - 0.2)
-			-- (0 + 0.2, -1 + 0.2)
-			-- (0 - 0.2, -1 + 0.2)
-			-- (0 - 0.2, -1 - 0.2);
-	\end{tikzpicture}
-	\hfill
-\end{center}
-
-
-\problem{}
-How many pairity bits does each message bit affect? \par
 Does this correlate with that message bit's index?
 \vfill
 
@@ -519,12 +567,6 @@ Does this correlate with that message bit's index?
 Say we have a message with exactly one single-bit error. \par
 If we know which parity bits are inconsistent, how can we find where the error is?
 
-\vfill
-\pagebreak
-
-\problem{}
-How efficient is the 16-bit hamming code?
-
 \vfill
 
 \problem{}<generalize-hamming>
@@ -532,10 +574,4 @@ Can you generalize this system for messages of 4, 64, or 256 bits?
 
 \vfill
 
-\problem{}
-How efficient is each code in \ref{generalize-hamming}? \par
-What do we sacrifice for this efficiency gain?
-
-\vfill
-
 \pagebreak