the properties of an ICSIE such as the property of redundant links and the relationship between the conventional network code with error-free links and an NCLE.

The paper is organized as follows. Several definitions, notations, and problem settings are given in Section \[Preliminary\]. The main results on equivalence relationships between an NCLE and an ICSIE for both a given network coding instance and a given index coding instance are derived in Section \[Equivalence\]. In Section \[Properties\], several properties of an NCLE are derived from those of an ICSIE based on the equivalence between an NCLE and an ICSIE. Finally, conclusions are presented in Section \[Sec:Conclusion\].

Preliminary {#Preliminary}
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In this section, we define network codes with link errors and index codes with side information errors and then state their problem settings and notations, where hatted notations are used for index coding to avoid confusion.

Notations
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Some of the notations are defined as follows:

1.  $Z[n]$ denotes a set of positive integers $\{1,2,...,n\}$.

2.  Let $\mathbb{F}_q$ be the finite field of size $q$, where $q$ is a power of prime and $\mathbb{F}_q^{*}=\mathbb{F}_q\setminus\{0\}$.

3.  For the vector $\bold{X}\in \mathbb{F}_q^n$, $\rm{\rm{wt}}(\bold{X})$ denotes Hamming weight of $\bold{X}$.

4.  Let $\bold{X}_D$ be a sub-vector 