$u$ is a source node, incoming symbols of $u$ denote the source messages possessed by $u$, meaning that up to $\delta_{e_1}$ message symbols are erroneous. Similarly, the decoding function $D_t$ of a terminal $t\in T$ is said to have an error resistance capability $\delta_t$ if $D_t$ can correctly obtain a decoded vector $\bold{X}_{\mathcal{F}(t)}$ from incoming symbols with less than or equal to $\delta_t$ symbol errors. In such a case, a network code with link errors is summarized as follows.

Let $\bm{\delta}=(\delta_{e_1},...,\delta_{e_{|E|}},\delta_{t_1},...,\delta_{t_{|T|}})$ be a vector whose elements correspond to the error resistance capability for each outgoing link and terminal in $E\cup T$. Then, a network code with link errors with parameters $(\bm{\delta},\mathbb{G})$ over $\mathbb{F}_q$, denoted by a $(\bm{\delta},\mathbb{G})$-NCLE consists of:

1.  An encoding function $F_e: \mathbb{F}_q^{|{\rm In}(e)|}\rightarrow \mathbb{F}_q$ for $e\in E$

2.  A decoding function $D_t: \mathbb{F}_q^{|{\rm In}(t)|}\rightarrow \mathbb{F}_q^{|\mathcal{F}(t)|}$ for $t\in T$

3.  Satisfying $F_e(\bold{X}_{{\rm In}(e)})=F_e(\bold{\tilde{X}}_{{\rm In}(e)})$ and $D_t(\bold{X}_{{\rm In}(t)})=D_t(\bold{\tilde{X}}_{{\rm In}(t)})$ for any $e\in E$ and $t\in T$, where ${\rm wt}(\bold{X}_{{\rm In}(e)}-\bold{\tilde{X}}_{{\rm In}(e)})\leq\delta_e$ and ${\rm{wt}}(\bold{X}_{{\rm In}(t)}-\bold{\tilde{X}}_{{\rm In}(t)})\leq\delta_t$.

Note that the error resistance capabilities are defined for encoding and decoding functions, that is, encoding functions for the outgoing links of one node can have different error resistance capabilities despite the fact that 