Channel decoder ($g:{\mathcal{Y}}^n\to \{1,\dots ,M\}$)

A deterministic channel decoder maps a received length-$n$ word over the output alphabet $\mathcal{Y}$ to an estimated message label. Given message $m$, successful decoding means $g(Y^n)=m$, where $Y^n$ is the random received word after the communication channel acts on the channel encoder's word $f(m)$. The channel decoder can recover a message despite changed symbols because it uses the relationships among complete codewords. The Shannon second coding theorem concerns choosing both maps so that the decoding-error probability becomes small at a specified rate.