State Shannon's noisy coding theorem for a binary symmetric channel, defining the terms involved.

Suppose a channel matrix, with output alphabet of size $n$, is such that the entries in each row are the elements of the set $\left\{p_{1}, \ldots, p_{n}\right\}$ in some order. Further suppose that all columns are permutations of one another. Show that the channel's information capacity $C$ is given by

$$C=\log n+\sum_{i=1}^{n} p_{i} \log p_{i}$$

Show that the information capacity of the channel matrix

$$\left(\begin{array}{cccc} \frac{1}{3} & \frac{1}{3} & \frac{1}{6} & \frac{1}{6} \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{3} & \frac{1}{3} \end{array}\right)$$

is given by $C=\frac{5}{3}-\log 3$.