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Unformatted text preview: he receiver can’t detect the
difference between receiving w1 with a unfortunately placed bit error and receiving w2 .
In this case, we cannot guarantee that all single bit errors will be corrected if we choose a
code where w1 and w2 are both valid code words.
What happens if we increase the Hamming distance between any two valid code
words to at least 2? More formally, let’s restrict ourselves to only sending some subset
S = {w1 , w2 , ..., ws } of the 2n possible words such that
HD(wi , wj ) ≥ 2 for all wi , wj ∈ S where i = j (6.3) Thus if the transmission of wi is corrupted by a single error, the result is not an element
of S and hence can be detected as an erroneous reception by the receiver, which knows
which messages are elements of S . A simple example is shown in Figure 62: 00 and 11 are
valid code words, and the receptions 01 and 10 are surely erroneous.
It should be easy to see what happens as we use a code whose minimum Hamming
distance between any two valid code words is D. We state the property formally:
Theorem 6.1 A code with a minimum Hamming distance of D can detect any error pattern of
D − 1 or fewer errors. Moreover, there is at least one error pattern with D errors that cannot be
detected reliably.
Hen...
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This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.
 Fall '13
 HariBalakrishnan

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