ECC edits

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

@@ -1,6 +1,6 @@
 \section{Hamming Codes}
 
-Say we have a 16-bit message, for example \texttt{1011 0101 1101 1001}. \par
+Say we have an $n$-bit message, for example \texttt{1011 0101 1101 1001}. \par
 We will number its bits in binary, from left to right:
 
 
@@ -62,13 +62,14 @@ We will number its bits in binary, from left to right:
 
 
 \problem{}
-In this 16-bit message, how many message bits have an index with a one as the last digit? \par
+If we number the bits of a 16-bit message as described above, \par
+how many message bits have an index with a one as the last digit? \par
 (i.e, an index that looks like \texttt{***1})
 
 \vspace{2cm}
 
 \problem{}
-Say we number the bits in a 32-bit message as above. \par
+Now consider a 32-bit message. \par
 How many message bits have an index with a one as the $n^\text{th}$ digit? \par
 
 \vspace{2cm}
@@ -76,7 +77,10 @@ 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 parity bits. This will reduce the amount of information we can send, but will also improve our error-detection capabilities.
+Now, let's come up with 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 allow the receiver to detect errors in the received message.
 
 \vspace{1mm}
 
@@ -174,7 +178,8 @@ Can this coding scheme correct a single-bit error?
 \vfill
 \pagebreak
 
-We'll now add four more parity 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}: \par
+This error-detection scheme is called the \textit{Hamming code}.
 
 \begin{center}
 \begin{tikzpicture}[scale = 1.25]
@@ -246,7 +251,7 @@ Bits \texttt{0100} and \texttt{1000} work in the same way. \par
 When counting bits in binary numbers, go from right to left.}
 
 \problem{}
-Which message bits does each parity bit cover? \par
+Which message bits does each parity bit \say{cover}? \par
 In other words, which message bits affect the value of each parity bit? \par
 
 \vspace{1mm}
@@ -358,7 +363,7 @@ Analyze this coding scheme.
 \vfill
 
 \problem{}
-Each of the following messages has either 0, 1, or two errors. \par
+Each of the following messages has either one, two, or no errors. \par
 Find the errors and correct them if possible. \par
 \hint{Bit \texttt{0000} should tell you how many errors you have.}
 
@@ -590,7 +595,7 @@ A \textit{message stream} is an infinite string of binary digits.
 \problem{}
 Show that Hamming codes do not reliably detect bit deletions: \par
 \hint{
-	Create a 17-bit message whose first 16 bits are a valid Hamming block, \par
+	Create a 17-bit message whose first 16 bits are a valid Hamming-coded message, \par
 	and which is still valid when a bit (chosen by you; not the $17^\text{th}$) is deleted.
 }
 
@@ -598,10 +603,11 @@ Show that Hamming codes do not reliably detect bit deletions: \par
 
 \problem{}
 Convince yourself that Hamming codes cannot correct insertions. \par
-Then, create a 16-bit message that...
+Then, create a 16-bit message that:
 \begin{itemize}
-	\item is a valid Hamming block, and
-	\item incorrectly "corrects" a single bit error when it encounters an insertion error.
+	\item is itself a valid Hamming code, and
+	\item incorrectly "corrects" a single bit error when it encounters an insertion error. \par
+	You may choose where the insertion occurs.
 \end{itemize}
 
 \vfill