Definition (Source-Channel Fixed-Length Code)

Given a discrete source $\{V_{i}\}^\infty_{i=1}$ with alphabet $\mathcal{V}$ and a discrete channel $(\mathcal{X},\mathcal{Y},\{P_{Y^{n}|X^{n}}\}^\infty_{i=1})$ an $m$-to-$n$ source-channel fixed-length code $\mathcal{C}_{m,n}$ with rate $\frac{m}{n}$ source symbol/channel symbol is a pair of maps $(f_{sc},g_{sc})$: $$f_{sc}:\mathcal{V}^{m}\to\mathcal{X}^{n}\mbox{ (encoder)}$$and $$g_{sc}:\mathcal{X}^{m}\to\mathcal{V}^{n}\mbox{ (decoder)}$$

Definition (Average Probability of Error)

$$\begin{align*} P_e(\mathcal{C}_{m,n}):&=P(\hat V^{m}\not=V^{m})\\ &=\sum\limits_{v^{m}\in V^{m}}P_{V^{m}}(v^{m})\sum\limits_{y^{n}\in\mathcal{Y}^{n}:g_{sc}(y^{n}\not=v^{m})}P_{Y^{n}|X^{n}}(y^{n}|f_{sc}(v^{m})) \end{align*}$$