links.

2.  $S$ denotes a set of source messages, that is, $\bar{s}\in\bar{S}$ has some elements $s\in S$.

3.  $T\subseteq V$ denotes a set of terminal nodes in ${\mathbb{G}}$, where terminal nodes do not have outgoing links.

4.  $\mathcal{F}$ denotes a function of the terminal nodes in ${\mathbb{G}}$, which indicates a set of indices of each terminal’s desired messages.

5.  For a link $e=(u,v)\in E$, ${\rm In}(e)$ denotes a set of incoming links of $u$, where $u,v\in V$.

6.  In the case of $u\in \bar{S}$, ${\rm In}(e)$ denotes a set of messages that $u$ has and ${\rm In}(t)$ denotes a set of incoming links of $t$ for which $t\in T$.

7.  At the ends of the links, errors may occur due to transmissions through links, referred to as link errors and source nodes may have erroneous source symbols.

8.  $\bm{\delta}=(\delta_{e_1},...,\delta_{e_{|E|}},\delta_{t_1},...,\delta_{t_{|T|}})$ is a vector whose elements correspond to the error resistance capability for each outgoing link from the node and terminal in $E\cup T$.

9.  When it is straightforward, we regard $s$, $e$, and $t$ as some indices.

In this network coding instance, we assume the followings:

1.  Each message is one symbol in $\mathbb{F}_q$.

2.  Each link carries one symbol in 