Discrete memoryless channel

A discrete memoryless channel has finite input and output alphabets and transition probabilities $P(y\mid x)$. Conditional on the input symbols, different channel uses have independent outputs governed by the same transition probabilities.

A finite-alphabet erasure channel transmits the input symbol unchanged with probability $1-f$ and replaces it by a distinct erasure flag with probability $f$, independently of the symbol. Given the flag, the posterior remains the input distribution; otherwise the input is known. Thus its mutual information is $(1-f)H(X)$.

For a finite-alphabet erasure channel with $k$ input symbols, $I(X;Y)=(1-f)H(X)$. Maximizing Shannon entropy at the uniform input gives channel capacity $C=(1-f){\log }_{2}k$ bits per use, by the Shannon second coding theorem.

A binary erasure channel transmits each input bit correctly with probability $1-\epsilon$ and replaces it by an erasure symbol with probability $\epsilon$. Its channel capacity is $1-\epsilon$ bits per use.

A Z-channel transmits one binary input without error while the other can change into the first. Unlike a binary symmetric channel, its errors are one-sided.