 Discrete memoryless channel

A discrete memoryless channel models transmission through finite input and output alphabets using fixed conditional probabilities $W(y\mid x)$. For each input $x$, these nonnegative numbers sum to one over the output symbols. Discrete refers to the symbols, and memoryless means that the noise of one use does not remember earlier uses.

For a fixed input word $x^n=(x_1,\dots ,x_n)$ sent without feedback, the output law isThis is conditional independence of the outputs given the prescribed input word. It does not require an encoder to choose independent input symbols: the codewords of a block code deliberately impose relationships among them. Those relationships let a decoder distinguish possible messages after noise.

The binary symmetric channel and binary erasure channel make two different memoryless noise models with a binary input. One hides flipped bits; the other marks missing bits. Their different channel capacities show why the whole transition matrix matters, as the flips versus erasures in a binary channel demonstrate.