Given $(n,M)$ code $\mathcal{C}_{n}$ with encoder/decoder pair $(f,g)$. ## Encoder To transmit message $w\in\mathcal{M}$, the encoder generates an (error-correcting) codeword $f(w)\in\mathcal{X}^{n}$ and sends it over the channel (by using the channel $n$ times).

Decoder

At channel output, $y^{n}\in\mathcal{Y}^{n}$ is received (which is, in general, a corrupted version of the sent codeword $f(w)$) and the decoder estimates the transmitted message as $$\hat w=g(y^{n})$$

The message $W$ is assumed to be a uniformly distributed $RV$ over its alphabet $\mathcal{M}=\{1,\cdots,M\}$ (as it typically represents a compressed data source which is assumed to be optimally compressed).

Therefor the pmf of $W$ is given by $$P_{W}(w):=P(W=w)=\frac{1}{M}, \ w\in\mathcal{M}$$