Channel code ($c:\{1,\dots ,m\}\to {\mathcal{X}}^n,d:{\mathcal{Y}}^n\to \{1,\dots ,m\}$)

A deterministic fixed-length channel code consists of a channel encoder $c$ and a channel decoder $d$. It represents $m$ possible messages by length-$n$ input codewords for a communication channel, and estimates the message from the resulting output word. Its code rate is $({\log }_{2}m)/n$ bits per channel use. The maximal decoding error probability requires reliable recovery of every message, while an average error criterion depends on the chosen message probability distribution.