Binary Symmetric Channel — Shannon Capacity

The binary symmetric channel (BSC) flips each bit independently with probability p. Shannon's channel capacity theorem states: C = 1 − H(p) = 1 − [−p·log₂p − (1−p)·log₂(1−p)] bits/use. This is the maximum error-free communication rate achievable by any code, no matter how complex.

Error probability p: 0.10

Code rate R: 0.50

Block length n: 100

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Shannon's Noisy Channel Theorem
If R < C: P_e → 0 as n → ∞
If R > C: P_e → 1 as n → ∞
H(p) = −p log₂p−(1−p)log₂(1−p)
C(BSC) = 1 − H(p) bits/channel use
Symmetric: C is max at p=0 or p=1