given as in Fig. \[fig:network example\]. A network code for Fig. \[fig:fig1\] can be described as follows:

1.  $X_{e_1}=X_{e_2}=X_{e_5}=X_{e_8}=X_{e_{10}}=X_{s_1}$

2.  $X_{e_3}=X_{e_4}=X_{e_7}=X_{e_{12}}=X_{s_2}$

3.  $X_{e_6}=X_{e_9}=X_{e_{11}}=X_{s_1}+X_{s_2}$

4.  $D_t$ is given in Algorithm 1.

Then, the corresponding index code for Fig. \[fig:fig7\] can be described as follows:

1.  The transmitted codeword $\hat{F}(\bold{\hat{X}})=\hat{{\bold{X}}}_B=(\hat{X}_B(e) : e\in E)$ consists of 12 components.

2.  $\hat{X}_B(e_i)=\hat{X}_{e_i}+\hat{X}_{s_1}$ for $i\in \{1, 2, 5, 8, 10\}$

3.  $\hat{X}_B(e_i)=\hat{X}_{e_i}+\hat{X}_{s_2}$ for $i\in \{3, 4, 7, 12\}$

4.  $\hat{X}_B(e_i)=\hat{X}_{e_i}+\hat{X}_{s_1}+\hat{X}_{s_2}$ for $i\in \{6, 9, 11\}$

5.  The decoding functions of receivers can be defined as in Theorem \[theorem:equ\].

Thus, by finding a network code for a given network coding instance, we can find the corresponding index code. Furthermore, if we find an index code with codelength $|E|$ for Fig. \[fig:fig7\], we can obtain the corresponding network code for Fig. \[fig:fig1\].

Equivalence for a Given Index Coding Instance
---------------------------------------------

Similarly, for a given index coding instance, we can construct the corresponding network coding instance. Since some index coding instances cannot be converted to corresponding network coding instances, as noted in Proposition $\ref{prop:condition}$, it is necessary to modify a given index coding instance to use the previous relationship between two coding instances. Specifically, some receivers and messages are 