\section{Hamming Codes}

Say we have a 16-bit message, for example \texttt{1011 0101 1101 1001}. \par
We will number its bits in binary, from left to right:


\begin{center}
\begin{tikzpicture}

	\node[anchor=west] at (-1.75, 0) {Bit};
	\node[anchor=west] at (-1.75, -0.5) {Index};

	\node at (0, 0) {\texttt{1}};
	\node at (1, 0) {\texttt{0}};
	\node at (2, 0) {\texttt{1}};
	\node at (3, 0) {\texttt{1}};
	\node at (4, 0) {\texttt{0}};
	\node at (5, 0) {\texttt{1}};
	\node at (6, 0) {\texttt{0}};
	\node at (7, 0) {\texttt{1}};

	\draw (-1.75, 0.25) -- (6.9, 0.25);
	\draw (-1.75, -0.25) -- (6.9, -0.25);
	\draw (-1.75, -0.75) -- (6.9, -0.75);

	\foreach \x in {-1.75,-0.5,0.5,...,6.5} {
		\draw (\x, 0.25) -- (\x, -0.75);
	}

	\node[color=gray] at (0, -0.5) {\texttt{0000}};
	\node[color=gray] at (1, -0.5) {\texttt{0001}};
	\node[color=gray] at (2, -0.5) {\texttt{0010}};
	\node[color=gray] at (3, -0.5) {\texttt{0011}};
	\node[color=gray] at (4, -0.5) {\texttt{0100}};
	\node[color=gray] at (5, -0.5) {\texttt{0101}};
	\node[color=gray] at (6, -0.5) {\texttt{0110}};
	\node[color=gray] at (7, -0.5) {\texttt{0111}};


	\draw[fill = white, draw = none]
		(6.9, 0.25)
		-- (7.1, 0)
		-- (6.9, -0.25)
		-- (7.1, -0.5)
		-- (6.9, -0.75)
		-- (7.5, -0.75)
		-- (7.5, 0.25)
	;

	\draw (6.9, 0.25)
		-- (7.1, 0)
		-- (6.9, -0.25)
		-- (7.1, -0.5)
		-- (6.9, -0.75)
	;


	\node[anchor=west,color=gray] at (7.2, -0.25) { and so on...};

\end{tikzpicture}
\end{center}


\problem{}
In this 16-bit message, 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
How many message bits have an index with a one as the $n^\text{th}$ digit? \par

\vspace{2cm}




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.

\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.

\begin{center}
\begin{tikzpicture}[scale = 1.25]
	\node at (0.0, 0) {\texttt{X}};
	\node at (0.5, 0) {\texttt{0}};
	\node at (1.0, 0) {\texttt{1}};
	\node at (1.5, 0) {\texttt{1}};

	\node at (0.0, -0.5) {\texttt{0}};
	\node at (0.5, -0.5) {\texttt{1}};
	\node at (1.0, -0.5) {\texttt{0}};
	\node at (1.5, -0.5) {\texttt{1}};

	\node at (0.0, -1) {\texttt{1}};
	\node at (0.5, -1) {\texttt{1}};
	\node at (1.0, -1) {\texttt{0}};
	\node at (1.5, -1) {\texttt{1}};

	\node at (0.0, -1.5) {\texttt{1}};
	\node at (0.5, -1.5) {\texttt{0}};
	\node at (1.0, -1.5) {\texttt{0}};
	\node at (1.5, -1.5) {\texttt{1}};

	\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 (-0.2,-0.2) -- (0.2, -0.2) -- (0.2, 0.2) -- (-0.2, 0.2) -- (-0.2,-0.2);
\end{tikzpicture}
\end{center}

\problem{}
What is the value of the pairity bit in the message above?

\vfill

\problem{}
Can this coding scheme detect a transposition error? \par
Can this coding scheme detect two single-bit errors? \par
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}:

\begin{center}
\begin{tikzpicture}[scale = 1.25]
	\node at (0.0, 0) {\texttt{X}};
	\node at (0.5, 0) {\texttt{X}};
	\node at (1.0, 0) {\texttt{X}};
	\node at (1.5, 0) {\texttt{1}};

	\node at (0.0, -0.5) {\texttt{X}};
	\node at (0.5, -0.5) {\texttt{1}};
	\node at (1.0, -0.5) {\texttt{0}};
	\node at (1.5, -0.5) {\texttt{1}};

	\node at (0.0, -1) {\texttt{X}};
	\node at (0.5, -1) {\texttt{1}};
	\node at (1.0, -1) {\texttt{0}};
	\node at (1.5, -1) {\texttt{1}};

	\node at (0.0, -1.5) {\texttt{1}};
	\node at (0.5, -1.5) {\texttt{0}};
	\node at (1.0, -1.5) {\texttt{0}};
	\node at (1.5, -1.5) {\texttt{1}};

	\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 (0 - 0.2, 0 - 0.2)
		-- (0 + 0.2, 0 - 0.2)
		-- (0 + 0.2, 0 + 0.2)
		-- (0 - 0.2, 0 + 0.2)
		-- (0 - 0.2, 0 - 0.2);
	\draw (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);
	\draw (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);
	\draw (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);
	\draw (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}
\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{0010} will count the pairity 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.

\problem{}
Compute all pairity bits in the message above.

\vfill

\problem{}
Each of the following messages has either 0, 1, or two errors. \par
Find the errors and correct them if possible.

\begin{center}
\hfill
\begin{tikzpicture}[scale = 1.25]
	\node at (0.0, 0) {\texttt{0}};
	\node at (0.5, 0) {\texttt{1}};
	\node at (1.0, 0) {\texttt{1}};
	\node at (1.5, 0) {\texttt{1}};

	\node at (0.0, -0.5) {\texttt{0}};
	\node at (0.5, -0.5) {\texttt{1}};
	\node at (1.0, -0.5) {\texttt{1}};
	\node at (1.5, -0.5) {\texttt{1}};

	\node at (0.0, -1) {\texttt{0}};
	\node at (0.5, -1) {\texttt{0}};
	\node at (1.0, -1) {\texttt{1}};
	\node at (1.5, -1) {\texttt{1}};

	\node at (0.0, -1.5) {\texttt{1}};
	\node at (0.5, -1.5) {\texttt{1}};
	\node at (1.0, -1.5) {\texttt{1}};
	\node at (1.5, -1.5) {\texttt{0}};

	\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 (0 - 0.2, 0 - 0.2)
		-- (0 + 0.2, 0 - 0.2)
		-- (0 + 0.2, 0 + 0.2)
		-- (0 - 0.2, 0 + 0.2)
		-- (0 - 0.2, 0 - 0.2);
	\draw (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);
	\draw (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);
	\draw (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);
	\draw (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
\begin{tikzpicture}[scale = 1.25]
	\node at (0.0, 0) {\texttt{1}};
	\node at (0.5, 0) {\texttt{1}};
	\node at (1.0, 0) {\texttt{0}};
	\node at (1.5, 0) {\texttt{1}};

	\node at (0.0, -0.5) {\texttt{1}};
	\node at (0.5, -0.5) {\texttt{0}};
	\node at (1.0, -0.5) {\texttt{1}};
	\node at (1.5, -0.5) {\texttt{0}};

	\node at (0.0, -1) {\texttt{0}};
	\node at (0.5, -1) {\texttt{1}};
	\node at (1.0, -1) {\texttt{1}};
	\node at (1.5, -1) {\texttt{0}};

	\node at (0.0, -1.5) {\texttt{1}};
	\node at (0.5, -1.5) {\texttt{1}};
	\node at (1.0, -1.5) {\texttt{0}};
	\node at (1.5, -1.5) {\texttt{1}};

	\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 (0 - 0.2, 0 - 0.2)
		-- (0 + 0.2, 0 - 0.2)
		-- (0 + 0.2, 0 + 0.2)
		-- (0 - 0.2, 0 + 0.2)
		-- (0 - 0.2, 0 - 0.2);
	\draw (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);
	\draw (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);
	\draw (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);
	\draw (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
\begin{tikzpicture}[scale = 1.25]
	\node at (0.0, 0) {\texttt{0}};
	\node at (0.5, 0) {\texttt{1}};
	\node at (1.0, 0) {\texttt{1}};
	\node at (1.5, 0) {\texttt{1}};

	\node at (0.0, -0.5) {\texttt{1}};
	\node at (0.5, -0.5) {\texttt{0}};
	\node at (1.0, -0.5) {\texttt{1}};
	\node at (1.5, -0.5) {\texttt{1}};

	\node at (0.0, -1) {\texttt{1}};
	\node at (0.5, -1) {\texttt{0}};
	\node at (1.0, -1) {\texttt{1}};
	\node at (1.5, -1) {\texttt{1}};

	\node at (0.0, -1.5) {\texttt{1}};
	\node at (0.5, -1.5) {\texttt{0}};
	\node at (1.0, -1.5) {\texttt{0}};
	\node at (1.5, -1.5) {\texttt{0}};

	\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 (0 - 0.2, 0 - 0.2)
		-- (0 + 0.2, 0 - 0.2)
		-- (0 + 0.2, 0 + 0.2)
		-- (0 - 0.2, 0 + 0.2)
		-- (0 - 0.2, 0 - 0.2);
	\draw (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);
	\draw (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);
	\draw (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);
	\draw (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}

\begin{solution}
	\textbf{1:} Single error at position \texttt{1010} \par
	\textbf{2:} Double error \par
	\textbf{3:} No error \par
\end{solution}


\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
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


\problem{}
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>
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