541 lines
14 KiB
TeX
541 lines
14 KiB
TeX
\section{Hamming Codes}
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Say we have a 16-bit message, for example \texttt{1011 0101 1101 1001}. \par
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We will number its bits in binary, from left to right:
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\begin{center}
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\begin{tikzpicture}
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\node[anchor=west] at (-1.75, 0) {Bit};
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\node[anchor=west] at (-1.75, -0.5) {Index};
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\node at (0, 0) {\texttt{1}};
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\node at (1, 0) {\texttt{0}};
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\node at (2, 0) {\texttt{1}};
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\node at (3, 0) {\texttt{1}};
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\node at (4, 0) {\texttt{0}};
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\node at (5, 0) {\texttt{1}};
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\node at (6, 0) {\texttt{0}};
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\node at (7, 0) {\texttt{1}};
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\draw (-1.75, 0.25) -- (6.9, 0.25);
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\draw (-1.75, -0.25) -- (6.9, -0.25);
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\draw (-1.75, -0.75) -- (6.9, -0.75);
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\foreach \x in {-1.75,-0.5,0.5,...,6.5} {
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\draw (\x, 0.25) -- (\x, -0.75);
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}
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\node[color=gray] at (0, -0.5) {\texttt{0000}};
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\node[color=gray] at (1, -0.5) {\texttt{0001}};
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\node[color=gray] at (2, -0.5) {\texttt{0010}};
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\node[color=gray] at (3, -0.5) {\texttt{0011}};
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\node[color=gray] at (4, -0.5) {\texttt{0100}};
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\node[color=gray] at (5, -0.5) {\texttt{0101}};
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\node[color=gray] at (6, -0.5) {\texttt{0110}};
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\node[color=gray] at (7, -0.5) {\texttt{0111}};
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\draw[fill = white, draw = none]
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(6.9, 0.25)
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-- (7.1, 0)
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-- (6.9, -0.25)
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-- (7.1, -0.5)
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-- (6.9, -0.75)
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-- (7.5, -0.75)
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-- (7.5, 0.25)
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;
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\draw (6.9, 0.25)
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-- (7.1, 0)
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-- (6.9, -0.25)
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-- (7.1, -0.5)
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-- (6.9, -0.75)
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;
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\node[anchor=west,color=gray] at (7.2, -0.25) { and so on...};
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\end{tikzpicture}
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\end{center}
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\problem{}
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In this 16-bit message, how many message bits have an index with a one as the last digit? \par
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(i.e, an index that looks like \texttt{***1})
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\vspace{2cm}
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\problem{}
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Say we number the bits in a 32-bit message as above. \par
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How many message bits have an index with a one as the $n^\text{th}$ digit? \par
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\vspace{2cm}
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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.
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\vspace{1mm}
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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.
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\begin{center}
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\begin{tikzpicture}[scale = 1.25]
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\node at (0.0, 0) {\texttt{X}};
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\node at (0.5, 0) {\texttt{0}};
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\node at (1.0, 0) {\texttt{1}};
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\node at (1.5, 0) {\texttt{1}};
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\node at (0.0, -0.5) {\texttt{0}};
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\node at (0.5, -0.5) {\texttt{1}};
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\node at (1.0, -0.5) {\texttt{0}};
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\node at (1.5, -0.5) {\texttt{1}};
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\node at (0.0, -1) {\texttt{1}};
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\node at (0.5, -1) {\texttt{1}};
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\node at (1.0, -1) {\texttt{0}};
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\node at (1.5, -1) {\texttt{1}};
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\node at (0.0, -1.5) {\texttt{1}};
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\node at (0.5, -1.5) {\texttt{0}};
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\node at (1.0, -1.5) {\texttt{0}};
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\node at (1.5, -1.5) {\texttt{1}};
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\draw (-0.25, 0.25) -- (1.75, 0.25);
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\draw (-0.25, -0.25) -- (1.75, -0.25);
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\draw (-0.25, -0.75) -- (1.75, -0.75);
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\draw (-0.25, -1.25) -- (1.75, -1.25);
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\draw (-0.25, -1.75) -- (1.75, -1.75);
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\draw (-0.25, 0.25) -- (-0.25, -1.75);
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\draw (0.25, 0.25) -- (0.25, -1.75);
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\draw (0.75, 0.25) -- (0.75, -1.75);
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\draw (1.25, 0.25) -- (1.25, -1.75);
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\draw (1.75, 0.25) -- (1.75, -1.75);
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\draw (-0.2,-0.2) -- (0.2, -0.2) -- (0.2, 0.2) -- (-0.2, 0.2) -- (-0.2,-0.2);
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\end{tikzpicture}
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\end{center}
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\problem{}
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What is the value of the pairity bit in the message above?
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\vfill
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\problem{}
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Can this coding scheme detect a transposition error? \par
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Can this coding scheme detect two single-bit errors? \par
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Can this coding scheme correct a single-bit error?
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\vfill
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\pagebreak
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We'll now add four more pairity bits, in positions \texttt{0001}, \texttt{0010}, \texttt{0100}, and \texttt{1000}:
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\begin{center}
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\begin{tikzpicture}[scale = 1.25]
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\node at (0.0, 0) {\texttt{X}};
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\node at (0.5, 0) {\texttt{X}};
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\node at (1.0, 0) {\texttt{X}};
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\node at (1.5, 0) {\texttt{1}};
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\node at (0.0, -0.5) {\texttt{X}};
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\node at (0.5, -0.5) {\texttt{1}};
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\node at (1.0, -0.5) {\texttt{0}};
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\node at (1.5, -0.5) {\texttt{1}};
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\node at (0.0, -1) {\texttt{X}};
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\node at (0.5, -1) {\texttt{1}};
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\node at (1.0, -1) {\texttt{0}};
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\node at (1.5, -1) {\texttt{1}};
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\node at (0.0, -1.5) {\texttt{1}};
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\node at (0.5, -1.5) {\texttt{0}};
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\node at (1.0, -1.5) {\texttt{0}};
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\node at (1.5, -1.5) {\texttt{1}};
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\draw (-0.25, 0.25) -- (1.75, 0.25);
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\draw (-0.25, -0.25) -- (1.75, -0.25);
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\draw (-0.25, -0.75) -- (1.75, -0.75);
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\draw (-0.25, -1.25) -- (1.75, -1.25);
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\draw (-0.25, -1.75) -- (1.75, -1.75);
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\draw (-0.25, 0.25) -- (-0.25, -1.75);
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\draw (0.25, 0.25) -- (0.25, -1.75);
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\draw (0.75, 0.25) -- (0.75, -1.75);
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\draw (1.25, 0.25) -- (1.25, -1.75);
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\draw (1.75, 0.25) -- (1.75, -1.75);
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\draw (0 - 0.2, 0 - 0.2)
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-- (0 + 0.2, 0 - 0.2)
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-- (0 + 0.2, 0 + 0.2)
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-- (0 - 0.2, 0 + 0.2)
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-- (0 - 0.2, 0 - 0.2);
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\draw (0.5 - 0.2, 0 - 0.2)
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-- (0.5 + 0.2, 0 - 0.2)
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-- (0.5 + 0.2, 0 + 0.2)
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-- (0.5 - 0.2, 0 + 0.2)
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-- (0.5 - 0.2, 0 - 0.2);
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\draw (1 - 0.2, 0 - 0.2)
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-- (1 + 0.2, 0 - 0.2)
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-- (1 + 0.2, 0 + 0.2)
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-- (1 - 0.2, 0 + 0.2)
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-- (1 - 0.2, 0 - 0.2);
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\draw (0 - 0.2, -0.5 - 0.2)
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-- (0 + 0.2, -0.5 - 0.2)
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-- (0 + 0.2, -0.5 + 0.2)
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-- (0 - 0.2, -0.5 + 0.2)
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-- (0 - 0.2, -0.5 - 0.2);
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\draw (0 - 0.2, -1 - 0.2)
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-- (0 + 0.2, -1 - 0.2)
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-- (0 + 0.2, -1 + 0.2)
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-- (0 - 0.2, -1 + 0.2)
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-- (0 - 0.2, -1 - 0.2);
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\end{tikzpicture}
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\end{center}
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Bit \texttt{0001} will count the pairity of all bits with a one in the first digit of their index. \par
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Bit \texttt{0010} will count the pairity of all bits with a one in the second digit of their index. \par
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Bits \texttt{0100} and \texttt{1000} work in the same way.
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\problem{}
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Compute all pairity bits in the message above.
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\vfill
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\problem{}
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Each of the following messages has either 0, 1, or two errors. \par
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Find the errors and correct them if possible.
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\begin{center}
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\hfill
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\begin{tikzpicture}[scale = 1.25]
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\node at (0.0, 0) {\texttt{0}};
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\node at (0.5, 0) {\texttt{1}};
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\node at (1.0, 0) {\texttt{1}};
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\node at (1.5, 0) {\texttt{1}};
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\node at (0.0, -0.5) {\texttt{0}};
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\node at (0.5, -0.5) {\texttt{1}};
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\node at (1.0, -0.5) {\texttt{1}};
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\node at (1.5, -0.5) {\texttt{1}};
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\node at (0.0, -1) {\texttt{0}};
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\node at (0.5, -1) {\texttt{0}};
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\node at (1.0, -1) {\texttt{1}};
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\node at (1.5, -1) {\texttt{1}};
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\node at (0.0, -1.5) {\texttt{1}};
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\node at (0.5, -1.5) {\texttt{1}};
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\node at (1.0, -1.5) {\texttt{1}};
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\node at (1.5, -1.5) {\texttt{0}};
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\draw (-0.25, 0.25) -- (1.75, 0.25);
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\draw (-0.25, -0.25) -- (1.75, -0.25);
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\draw (-0.25, -0.75) -- (1.75, -0.75);
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\draw (-0.25, -1.25) -- (1.75, -1.25);
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\draw (-0.25, -1.75) -- (1.75, -1.75);
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\draw (-0.25, 0.25) -- (-0.25, -1.75);
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\draw (0.25, 0.25) -- (0.25, -1.75);
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\draw (0.75, 0.25) -- (0.75, -1.75);
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\draw (1.25, 0.25) -- (1.25, -1.75);
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\draw (1.75, 0.25) -- (1.75, -1.75);
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\draw (0 - 0.2, 0 - 0.2)
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-- (0 + 0.2, 0 - 0.2)
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-- (0 + 0.2, 0 + 0.2)
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-- (0 - 0.2, 0 + 0.2)
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-- (0 - 0.2, 0 - 0.2);
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\draw (0.5 - 0.2, 0 - 0.2)
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-- (0.5 + 0.2, 0 - 0.2)
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-- (0.5 + 0.2, 0 + 0.2)
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-- (0.5 - 0.2, 0 + 0.2)
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-- (0.5 - 0.2, 0 - 0.2);
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\draw (1 - 0.2, 0 - 0.2)
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-- (1 + 0.2, 0 - 0.2)
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-- (1 + 0.2, 0 + 0.2)
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-- (1 - 0.2, 0 + 0.2)
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-- (1 - 0.2, 0 - 0.2);
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\draw (0 - 0.2, -0.5 - 0.2)
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-- (0 + 0.2, -0.5 - 0.2)
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-- (0 + 0.2, -0.5 + 0.2)
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-- (0 - 0.2, -0.5 + 0.2)
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-- (0 - 0.2, -0.5 - 0.2);
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\draw (0 - 0.2, -1 - 0.2)
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-- (0 + 0.2, -1 - 0.2)
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-- (0 + 0.2, -1 + 0.2)
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-- (0 - 0.2, -1 + 0.2)
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-- (0 - 0.2, -1 - 0.2);
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\end{tikzpicture}
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\hfill
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\begin{tikzpicture}[scale = 1.25]
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\node at (0.0, 0) {\texttt{1}};
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\node at (0.5, 0) {\texttt{1}};
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\node at (1.0, 0) {\texttt{0}};
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\node at (1.5, 0) {\texttt{1}};
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\node at (0.0, -0.5) {\texttt{1}};
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\node at (0.5, -0.5) {\texttt{0}};
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\node at (1.0, -0.5) {\texttt{1}};
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\node at (1.5, -0.5) {\texttt{0}};
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\node at (0.0, -1) {\texttt{0}};
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\node at (0.5, -1) {\texttt{1}};
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\node at (1.0, -1) {\texttt{1}};
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\node at (1.5, -1) {\texttt{0}};
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\node at (0.0, -1.5) {\texttt{1}};
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\node at (0.5, -1.5) {\texttt{1}};
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\node at (1.0, -1.5) {\texttt{0}};
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\node at (1.5, -1.5) {\texttt{1}};
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\draw (-0.25, 0.25) -- (1.75, 0.25);
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\draw (-0.25, -0.25) -- (1.75, -0.25);
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\draw (-0.25, -0.75) -- (1.75, -0.75);
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\draw (-0.25, -1.25) -- (1.75, -1.25);
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\draw (-0.25, -1.75) -- (1.75, -1.75);
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|
|
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\draw (-0.25, 0.25) -- (-0.25, -1.75);
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\draw (0.25, 0.25) -- (0.25, -1.75);
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\draw (0.75, 0.25) -- (0.75, -1.75);
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\draw (1.25, 0.25) -- (1.25, -1.75);
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\draw (1.75, 0.25) -- (1.75, -1.75);
|
|
|
|
\draw (0 - 0.2, 0 - 0.2)
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-- (0 + 0.2, 0 - 0.2)
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|
-- (0 + 0.2, 0 + 0.2)
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-- (0 - 0.2, 0 + 0.2)
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-- (0 - 0.2, 0 - 0.2);
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\draw (0.5 - 0.2, 0 - 0.2)
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-- (0.5 + 0.2, 0 - 0.2)
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-- (0.5 + 0.2, 0 + 0.2)
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-- (0.5 - 0.2, 0 + 0.2)
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|
-- (0.5 - 0.2, 0 - 0.2);
|
|
\draw (1 - 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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|
-- (1 - 0.2, 0 - 0.2);
|
|
\draw (0 - 0.2, -0.5 - 0.2)
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|
-- (0 + 0.2, -0.5 - 0.2)
|
|
-- (0 + 0.2, -0.5 + 0.2)
|
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-- (0 - 0.2, -0.5 + 0.2)
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|
-- (0 - 0.2, -0.5 - 0.2);
|
|
\draw (0 - 0.2, -1 - 0.2)
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|
-- (0 + 0.2, -1 - 0.2)
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|
-- (0 + 0.2, -1 + 0.2)
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|
-- (0 - 0.2, -1 + 0.2)
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|
-- (0 - 0.2, -1 - 0.2);
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\end{tikzpicture}
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\hfill
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\begin{tikzpicture}[scale = 1.25]
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\node at (0.0, 0) {\texttt{0}};
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\node at (0.5, 0) {\texttt{1}};
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\node at (1.0, 0) {\texttt{1}};
|
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\node at (1.5, 0) {\texttt{1}};
|
|
|
|
\node at (0.0, -0.5) {\texttt{1}};
|
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\node at (0.5, -0.5) {\texttt{0}};
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\node at (1.0, -0.5) {\texttt{1}};
|
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\node at (1.5, -0.5) {\texttt{1}};
|
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\node at (0.0, -1) {\texttt{1}};
|
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\node at (0.5, -1) {\texttt{0}};
|
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\node at (1.0, -1) {\texttt{1}};
|
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\node at (1.5, -1) {\texttt{1}};
|
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|
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\node at (0.0, -1.5) {\texttt{1}};
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\node at (0.5, -1.5) {\texttt{0}};
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\node at (1.0, -1.5) {\texttt{0}};
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\node at (1.5, -1.5) {\texttt{0}};
|
|
|
|
\draw (-0.25, 0.25) -- (1.75, 0.25);
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|
\draw (-0.25, -0.25) -- (1.75, -0.25);
|
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\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);
|
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|
|
\draw (-0.25, 0.25) -- (-0.25, -1.75);
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\draw (0.25, 0.25) -- (0.25, -1.75);
|
|
\draw (0.75, 0.25) -- (0.75, -1.75);
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|
\draw (1.25, 0.25) -- (1.25, -1.75);
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|
\draw (1.75, 0.25) -- (1.75, -1.75);
|
|
|
|
\draw (0 - 0.2, 0 - 0.2)
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|
-- (0 + 0.2, 0 - 0.2)
|
|
-- (0 + 0.2, 0 + 0.2)
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-- (0 - 0.2, 0 + 0.2)
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|
-- (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}
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|
\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 |