Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 3I (Coding and Cryptography)

For a discrete memoryless channel with input $X$ and output $Y$, its information capacity iswhere $I(X;Y)$ is the mutual information and the maximum is over all input distributions. Capacity is measured in bits when logarithms have base two.

With rows indexed by outputs $A,B,\star$ and columns by inputs $A,B$, the channel matrix isThis is a binary erasure channel with erasure probability $1/2$. Conditional on no erasure, the input is recovered exactly, whereas an erasure supplies no information. Thuswhich is maximized by the uniform input distribution. Therefore

Let $q=\mathbb{P}(X=B)$. The merged channel is a Binary Z-channel: its second output occurs with probability $q/2$, while the output has one bit of uncertainty only when $X=B$. In terms of the binary entropy function $h_2$,At an interior maximum,so $(2-q)/q=4$ and $q=2/5$. Consequently