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