$\mathbb{F}_q$.

3.  $X_s$ denotes an element of a message vector $\bold{X}\in \mathbb{F}_q^{|S|}$.

4.  $X_e$ denotes a symbol on a link $e$ for which $e\in E$.

5.  For a set $A\subseteq S$, $\bold{X}_A$ denotes a sub-vector of $\bold{X}$ and for a set $B\subseteq E$, $\bold{X}_B$ denotes a vector consisting of $|B|$ symbols of the corresponding links.

6.  $\bold{\tilde{X}}_A=(\tilde{X}_1,...,\tilde{X}_{|A|})$ and $\bold{\tilde{X}}_B=(\tilde{X}_1,...,\tilde{X}_{|B|})$ denote vectors with erroneous symbol elements.

![Node processing in the network code.[]{data-label="fig:node processing"}](fig11)

Next, we describe node processing in the network code as in Fig. \[fig:node processing\], where $e^{\prime}_i, 1\leq i\leq l,$ denote the incoming edges of a node $u$ and $e_i, 1\leq i\leq k,$ denote the outgoing edges of $u$. At the node $u$, outgoing symbols for edges $e_i, 1\leq i\leq k,$ are computed by encoding functions as $X_{e_i}=F_{e_i}(\tilde{X}_{e^\prime_1},...,\tilde{X}_{e^\prime_l}), 1\leq i \leq k$. We consider a network code capable of resolving some link errors. Assume that there are less than or equal to $\delta_{e_1}$ symbol errors in the incoming links of $u$. If an encoding function $F_{e_1}$ can make a correct encoded outgoing symbol $X_{e_1}$ from $l$ incoming symbols with less than or equal to $\delta_{e_1}$ symbol errors, then $F_{e_1}$ is said to have an error resistance capability $\delta_{e_1}$. When 