 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-3/3g/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

R

n

codes can carry

M_{n}

messages with

n^{- 1} lo g_{2} M_{n}

tending to at least

R

C (p) = 1 - h_{2} (p), h_{2} (p) = - p lo g_{2} p - (1 - p) lo g_{2} (1 - p) .

For small positive

p

h_{2} (p) = p lo g_{2} \frac{1}{p} + \frac{p}{l n 2} + O (p^{2}),

C (p)

starts at one bit per channel use and decreases rapidly from one as errors are introduced.

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