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Unformatted text preview: ing codes. The
(7,4) Hamming code uses 3 parity bits to protect 4 data bits; 3 of the 4 data bits are involved
in each parity computation. The (15,11) Hamming code uses 4 parity bits to protect 11 data
bits, and 7 of the 11 data bits are used in each parity computation (these properties will
become apparent when we discuss the logic behind the construction of the Hamming code
in Section 6.4.4).
Looking at the diagrams, which show the data bits involved in each parity computation,
you should convince yourself that each possible single error (don’t forget errors in one of
the parity bits!) results in a unique combination of parity errors. Let’s work through the
argument for the (7,4) Hamming code. Here are the parity-check computations performed
by the receiver:
E1 = ( d 1 + d 2 + d 4 + p1 ) mod 2 E2 = ( d 1 + d 3 + d 4 + p2 ) mod 2 E3 = ( d 2 + d 3 + d 4 + p3 ) mod 2 where each Ei is called a syndrome bit because it helps the receiver diagnose the “illness”
(errors) in the received data. For each combination of syndr...
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This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.
- Fall '13