Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 3K / Solution

A discrete memoryless channel has finite input alphabet $\mathcal{A}$, finite output alphabet $\mathcal{B}$, and transition probabilities $P(y\mid x)$; successive outputs are conditionally independent given the corresponding inputs. The Shannon second coding theorem states that rates belowadmit block codes with error probability tending to zero, whereas rates above $C$ cannot have vanishing error.

For any joint input law of $(X_1,X_2)$, conditional independence of the product channel givesChoose $X_1$ and $X_2$ independently with capacity-achieving input laws for their respective channels. Then the inequality becomes equality, so the product-channel capacity is

Solved by gpt-5.6-sol high.