code with codelength $n-1$, that is, $(\hat{X}_1+\hat{X}_2,\hat{X}_2+\hat{X}_3,...,\hat{X}_{n-1}+\hat{X}_n)$. The necessity part is based on the fact that $\mathcal{I}(q,\mathcal{G},\bm{\delta}_s)$ is a set of all vectors in $\mathbb{F}_q^n$ except for $\bold{0}$ if $\mathcal{G}$ is $\bm{\delta}_s$-acyclic. Specifically, because $Z[n]$ is not a $\bm{\delta}_s$-cycle, we can assume that there is at least one receiver $R_1$ with a wanted message $\hat{X}_1$ and $|\mathcal{X}_1\cap Z[n]|\leq 2\delta_s^{(1)}$ without loss of generality. Then, every $\bold{\hat{Z}}\in\mathbb{F}_q^n$ with $\hat{Z}_1\neq 0$ is included in $\mathcal{I}(q,\mathcal{G},\bm{\delta}_s)$. Similarly, because $Z[n]\setminus\{1\}$ is not a $\bm{\delta}_s$-cycle, we can assume that there is at least one receiver $R_2$ with a wanted message $\hat{X}_2$ and $|\mathcal{X}_2\cap \{Z[n]\setminus\{1\}\}|\leq 2\delta_s^{(2)}$. Then, every $\bold{\hat{Z}}\in\mathbb{F}_q^n$ with $\hat{Z}_1=0$ and $\hat{Z}_2\neq 0$ is included in $\mathcal{I}(q,\mathcal{G},\bm{\delta}_s)$. The similar result for $R_i$ is that every $\bold{\hat{Z}}\in\mathbb{F}_q^n$ with $\hat{Z}_1=\hat{Z}_2=\cdots=\hat{Z}_{i-1}=0$ and $\hat{Z}_{i}\neq 0$ is included in $\mathcal{I}(q,\mathcal{G},\bm{\delta}_s)$, which means that $\mathcal{I}(q,\mathcal{G},\bm{\delta}_s)=\mathbb{F}_q^n\setminus\{\bold{0}\}$. In this case, all of the message vectors in $\mathbb{F}_q^n$ should be encoded to different codewords. Thus, $N_{\rm opt}^q(\bm{\delta}_s,\mathcal{G})=n$.

\[delta\_s-cycle\]

Equivalences Between Network Codes With Link Errors and Index Codes With Side Information Errors  {#Equivalence}
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In this section, we prove the equivalences between network codes with link errors and index codes with side information errors. First, their equivalence is proved for a given network coding 