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Unformatted text preview: ﬂat pieces), which
means that the decoding error probability decreases exponentially with the code rate. It is
also worth noting that the error probability is the same when c = 2 as when c = 2 − 1.
The reason, of course, is that the decoder obtains no additional information that it already
didn’t know from any 2 − 1 of the received bits.
Given a chunk of data of size s bits, we can now calculate the probability that it will
be in error after the error correction code has been applied. Each message word (1 bit in
the case of the replication code) will be decoded incorrectly with probability q , where q is
given by Eq.(6.2). The probability that the entire chunk of data will be decoded correctly is
given by (1 − q )s , and the desired error probability is therefore equal to 1 − (1 − q )s . When
q << 1, that error probability is approximately qs. This result should make intuitive sense.
Despite the exponential reduction in the probability of decoding error as c increases, SECTION 6.3. EMBEDDINGS AND HAMMING DI...
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 Fall '13
 HariBalakrishnan

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