Mathematical Statement
Theorem (Shannon, 1948):
- 1. For every discrete memoryless channel, the channel capacity
- has the following property. For any ε > 0 and R < C, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε.
- 2. If a probability of bit error pb is acceptable, rates up to R(pb) are achievable, where
- and is the binary entropy function
- 3. For any pb, rates greater than R(pb) are not achievable.
(MacKay (2003), p. 162; cf Gallager (1968), ch.5; Cover and Thomas (1991), p. 198; Shannon (1948) thm. 11)
Read more about this topic: Noisy-channel Coding Theorem
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