is a linear code. However, a general index code containing a nonlinear case is considered in this paper. Thus, we should modify and re-prove some of the properties of a $(\bm{\delta}_s,\mathcal{G})$-ICSIE.

Let $\mathcal{I}(q,\mathcal{G},\bm{\delta}_s)$ be a set of vectors defined by $$\mathcal{I}(q,\mathcal{G},\bm{\delta}_s)=\bigcup_{i\in Z[m]}\mathcal{I}_i(q,\mathcal{G},\delta_s^{(i)})
\nonumber$$ where $\mathcal{I}_i(q,\mathcal{G},\delta_s^{(i)})=\{\bold{\hat{Z}}\in\mathbb{F}_q^n | {\rm wt}(\bold{\hat{Z}}_{\mathcal{X}_i})\leq2\delta_s^{(i)}, \bold{\hat{Z}}_{f(i)}\neq\bold{0}\}$. Then, a property of a $(\bm{\delta}_s,\mathcal{G})$-ICSIE is given in the following theorem.

A $(\bm{\delta}_s,\mathcal{G})$-ICSIE is valid if and only if $$\hat{F}(\bold{\hat{X}})\neq \hat{F}(\bold{\hat{X}}^{\prime}) \textrm{ for all } \bold{\hat{X}}-\bold{\hat{X}}^{\prime}\in\mathcal{I}(q,\mathcal{G},\bm{\delta}_s).
\label{equ:encoding}$$ \[encoding\]

Each receiver $R_i$ has to recover $\bold{\hat{X}}_{f(i)}$ using the received codeword $\hat{F}(\bold{\hat{X}})$ and the side information $\bold{\hat{X}}_{\tilde{\mathcal{X}}_i}$. Then, the sender has to encode some confusing messages as different codewords. Because each receiver $R_i$ is only interested in $\bold{\hat{X}}_{f(i)}$, the codewords of distinct messages with an identical $\bold{\hat{X}}_{f(i)}$ do not need to be distinguished. Moreover, the codewords of two messages $\bold{\hat{X}}$ and $\bold{\hat{X}}^{\prime}$ such that ${\rm{wt}}(\bold{\hat{X}}_{\mathcal{X}_i}-\bold{\hat{X}}^{\prime}_{\mathcal{X}_i})>2\delta_s^{(i)}$ do not need to be distinguished because they can be distinguished by the side information of $R_i$. Thus, only problematic types of messages can be represented by two messages $\bold{\hat{X}}$ and $\bold{\hat{X}}^{\prime}$ such that $\bold{\hat{X}}_{f(i)}\neq\bold{\hat{X}}^{\prime}_{f(i)}$ and ${\rm{wt}}(\bold{\hat{X}}_{\mathcal{X}_i}-\bold{\hat{X}}^{\prime}_{\mathcal{X}_i})\leq2\delta_s^{(i)}$. Given that $\bold{\hat{X}}$ and $\bold{\hat{X}}^{\prime}$ are confusing, $\hat{F}(\bold{\hat{X}})\neq\hat{F}(\bold{\hat{X}}^{\prime})$ should be satisfied for all $i\in Z[m]$.

For a linear $(\bm{\delta}_s,\mathcal{G})$-ICSIE, (\[equ:encoding\]) 