2023-06-19 15:42:33 -07:00
\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} } ;
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-- (0.5 - 0.2, 0 - 0.2);
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\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
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\node at (0.0, 0) { \texttt { 0} } ;
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\node at (1.5, -1.5) { \texttt { 0} } ;
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\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} } ;
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-- (0 + 0.2, -1 + 0.2)
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-- (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} } ;
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-- (0.5 - 0.2, 0 - 0.2);
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-- (1 + 0.2, 0 - 0.2)
-- (1 + 0.2, 0 + 0.2)
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-- (1 - 0.2, 0 - 0.2);
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-- (0 - 0.2, -0.5 - 0.2);
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\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.
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\hfill
\begin { tikzpicture} [scale = 1.25]
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\hfill
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\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?
2023-06-20 10:07:35 -07:00
\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?
2023-06-19 15:42:33 -07:00
\vfill
\pagebreak
