= Channel decoder
{title2=$g:\mathcal Y^n\to\{1,\ldots,M\}$}
= Decoder
{synonym}
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.
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