# The Computability of the Channel Reliability Function and Related Bounds
## 6 Behavior of the Evurgation-Bound Rates  
In this section, we consider the behavior of the expurgation-bound rate. \(R_{ex}^{k}\) occurs in the expurgation bound as a lower bound for the channel reliability function, where \(k\) is the parameter for the \(k\)-letter description. Let \(\mathcal{X}_{1},\mathcal{Y}_{1},\mathcal{X}_{2},\mathcal{Y}_{2}\) be arbitrary finite non-trivial alphabets, and \(W_{l}\in CH(\mathcal{X}_{l},\mathcal{Y}_{l})\) for \(l=1,2\). We want to examine \(R_{ex}^{k}\). **Theorem 20**. : _There exist non-trivial alphabets \(\mathcal{X}_{1},\mathcal{Y}_{1},\mathcal{X}_{2},\mathcal{Y}_{2}\) and channels \(W_{l}\in CH(\mathcal{X}_{l},\mathcal{Y}_{l})\) for \(l=1,2\), such that for all \(k\), there exists \(k\geq k\) with_  
\[R_{ex}^{k}(W_{1}\otimes W_{2})\
eq R_{ex}^{k}(W_{1})+R_{ex}^{k}(W_{2}).\]  
Proof. : Assume that for all \(\mathcal{X}_{1},\mathcal{Y}_{1},\mathcal{X}_{2},\mathcal{Y}_{2}\) and \(W_{l}\in CH(\mathcal{X}_{l},\mathcal{Y}_{l})\) with \(l=1,2\) for all \(k\in\mathbb{N}\),  
\[R_{ex}^{k}(W_{1}\otimes W_{2})=R_{ex}^{k}(W_{1})+R_{ex}^{k}(W_{2}).\]  
We now take \(\mathcal{X}_{1}^{\prime},\mathcal{Y}_{1}^{\prime},\mathcal{X}_{2}^{\prime}, \mathcal{Y}_{2}^{\prime}\) such that \(C_{0}\) is superadditive. Then, we have for certain \(W_{1}^{\prime},W_{2}^{\prime}\) with \(W_{l}^{\prime}\in CH(\mathcal{X}_{l}^{\prime},\mathcal{Y}_{l}^{\prime})\),  
\[C_{0}(W_{1}^{\prime}\otimes W_{2}^{\prime})>C_{0}(W_{1}^{\prime})+C_{0}(W_{2}^{ \prime}). \tag{37}\]Then,  
\[C_{0}(W_{1}^{\prime}\otimes W_{2}^{\prime}) = \lim_{k\to\infty}R_{ex}^{k}(W_{1}^{\prime}\otimes W_{2}^{\prime})\] \[= \lim_{k\to\infty}R_{ex}^{k}(W_{1}^{\prime})+R_{ex}^{k}(W_{2}^{ \prime})\] \[= C_{0}(W_{1}^{\prime})+C_{0}(W_{2}^{\prime}).\]  
This is a contradiction, and thus the theorem is proven. We improve the statement of Theorem 20 with the following theorem. **Theorem 21**. : _There exist non-trivial alphabets \(\mathcal{X}_{1},\mathcal{Y}_{1},\mathcal{X}_{2},\mathcal{Y}_{2}\), channels \(W_{l}\in CH(\mathcal{X}_{l},\mathcal{Y}_{l})\) for \(l=1,2\) and a \(\hat{k}\), such that for all \(k\geq\hat{k}\),_  
\[R_{ex}^{k}(W_{1}\otimes W_{2})>R_{ex}^{k}(W_{1})+R_{ex}^{k}(W_{2})\]  
_holds true._  
Proof. : Assume the statement of the theorem is false, which means for all channels \(W_{l}\in CH(\mathcal{X}_{l},\mathcal{Y}_{l})\) with \(l=1,2\), the following applies: There exists a sequence \(\{k_{j}\}_{j\in\mathbb{N}}\subset\mathbb{N}\) with \(\lim_{j\to\infty}k_{j}=+\infty\), such that  
\[R_{ex}^{k_{l}}(W_{1}\otimes W_{2})\leq R_{ex}^{k_{l}}(W_{1})+R_{ex}^{k_{l}}(W_ {2})\]  
for \(l\in\mathbb{N}\). We now take \(\hat{\mathcal{X}}_{1},\hat{\mathcal{Y}}_{1},\hat{\mathcal{X}}_{2},\hat{ \mathcal{Y}}_{2}\) so that \(C_{0}\) is superadditive for these alphabets. Then, we have for certain \(\hat{W}_{1},\hat{W}_{2}\) with \(\hat{W}_{l}\in CH(\mathcal{X}_{l},\mathcal{Y}_{l})\) for \(l=1,2\),  
\[C_{0}(\hat{W}_{1}\otimes\hat{W}_{2})>C_{0}(\hat{W}_{1})+C_{0}(\hat{W}_{2}). \tag{38}\]  
Then,  
\[C_{0}(\hat{W}_{1}\otimes\hat{W}_{2})=\lim_{j\to\infty}R_{ex}^{k_{ l}}(\hat{W}_{1}\otimes\hat{W}_{2}) \leq \lim_{j\to\infty}\biggl{(}R_{ex}^{\hat{k}_{l}}(\hat{W}_{1})+R_{ex }^{k_{l}}(\hat{W}_{2})\biggr{)}\] \[R_{ex}^{k}(W_{1}\otimes W_{2})=R_{ex}^{k}(W_{1})+R_{ex}^{k}(W_{2}).\]0  
This is a contradiction to (38), and thus the theorem is proven. We have already observed that the function \(E(W,\cdot)\) exhibits significantly different behavior over certain rate intervals \([R,\hat{R}]\). In particular, we have analyzed the impact of the channel product \(W_{1}\otimes W_{2}\) on the intervals \((R_{\infty}(W_{1}\otimes W_{2}),C(W_{1}\otimes W_{2}))\) and \((E_{ex}^{k}(W_{1}\otimes W_{2}),C(W_{1}\otimes W_{2}))\) for \(k\in\mathbb{N}\). For the first interval, we established the relation  
\[R_{ex}^{k}(W_{1}\otimes W_{2})=R_{ex}^{k}(W_{1})+R_{ex}^{k}(W_{2}).\]1  
However, for the second interval, we have shown that such a simple additive behavior does not hold.