W {\displaystyle W} is the message to be transmitted; X {\displaystyle X} is the channel input symbol ( X n {\displaystyle X^{n}} is a sequence of n {\displaystyle n} symbols) taken in an alphabet X {\displaystyle {\mathcal {X}}} ; Y {\displaystyle Y} is the channel output symbol ( Y n {\displaystyle Y^{n}} is a sequence of n {\displaystyle n} symbols) taken in an alphabet Y {\displaystyle {\mathcal {Y}}} ; W ^ {\displaystyle {\hat {W}}} is the estimate of the transmitted message; f n {\displaystyle f_{n}} is the encoding function for a block of length n {\displaystyle n} ; p ( y | x ) = p Y | X ( y | x ) {\displaystyle p(y|x)=p_{Y|X}(y|x)} is the noisy channel, which is modeled by a conditional probability distribution; and, g n {\displaystyle g_{n}} is the decoding function for a block of length n {\displaystyle n} .