$(\bm{\delta}_s,\mathcal{G})$-ICSIE exists.

This can be proved by a method similar to that in [@EQU] but the differences are the existence of $\hat{t}_{\rm all}$ and the fact that there are link errors and side information errors.

Necessity: Assume that there exists a $(\bm{\delta},\mathbb{G})$-NCLE. First, the encoding function of the corresponding $(\bm{\delta}_s,\mathcal{G})$-ICSIE is defined as $\hat{F}(\bold{\hat{X}})=\bold{\hat{X}}_B=(\hat{X}_B(e): e\in E)$ such that $\hat{X}_B(e)=\hat{X}_e+\bar{F}_e(\hat{X}_1,...,\hat{X}_{|S|})$. Next, we define the decoding functions and show that all receivers in the corresponding index coding instance can recover what they want. It is already given that $R_e$ can be described as $\mathcal{X}_e={\rm In}(e)$, $f(e)=\{e\}$, and $\delta_s^{(e)}=\delta_e$. Thus, for each $e^{\prime}\in {\rm In}(e)$, the decoder can compute $\hat{X}_B(e^{\prime})-\hat{X}_{\tilde{e^{\prime}}}$, which can be an erroneous value of $\bar{F}_{e^{\prime}}(\hat{X}_1,...,\hat{X}_{|S|})$. Then, $\delta_s^{(e)}$ symbols of them can be erroneous. Since the link $e$ in the network coding instance has an error resistance capability $\delta_e=\delta_s^{(e)}$, evaluating these symbols with $F_e$ results in the correct value of $\bar{F}_e(\hat{X}_1,...,\hat{X}_{|S|})$. Now, $R_e$ can obtain $\hat{X}_e$ by subtracting $\bar{F}_e(\hat{X}_1,...,\hat{X}_{|S|})$ from $\hat{X}_B(e)$. It is also given that $R_t$ can be described as $\mathcal{X}_t={\rm In}(t)$, $f(t)=\mathcal{F}(t)$, and $\delta_s^{(t)}=\delta_t$. Similar to the $R_e$ case, $R_t$ can recover what it wants because it can obtain $\bar{F}_e(\hat{X}_1,...,\hat{X}_{|S|})$ for all $e\in {\rm In}(t)$, whose $\delta_s^{(t)}$ symbols can 