$W_{d_k,\mathcal{H}_i\backslash \{ k\}}, k\in \mathcal{H}_i$, is equal to the minimum cover of $\mathcal{H}_i$.\
As an example, consider the incidence matrix as shown in Fig. \[fig:conn\_matrix\], which corresponds to a system with $P=7$ servers and $K=5$ users, where each user connects to $\rho=4$ servers. Assume that the server storage capacity is $M_S=\frac{N}{\rho-2}$ and $t=1$. In this setting, coded subsegments of requested files can be delivered to $t+1=2$ users through multicasting, and it is sufficient for each user to receive coded segments from $\rho-2=2$ servers. Then, for the user set $\mathcal{H}_i=\{1,2\}$, we consider the submatrix corresponding to the columns $1$ and $2$ and rows $1$ and $2$ (marked by the blue dashed lines in Fig. \[fig:conn\_matrix\]), which is the smallest submatrix satisfying the condition that each column has at least $\rho-z=2$ $1$s. Hence, the minimum cover for $\mathcal{H}_i=\{ 1,2 \}$ is equal to the number of rows of this submatrix, that is, $2$. For $\mathcal{H}_i=\{3,4\}$ (marked by the red dashed lines in Fig. \[fig:conn\_matrix\]), the minimum cover is $3$. Thus, from , for segments $W_{d_k,\{3,4\} \backslash \{ k\}}, k\in \{3,4\}$, $S_3$ transmits the message $X_3(\{3,4\}) = \bigoplus_{k \in \{3,4\}} C_{d_k,\{3,4\} \backslash \{k\}}^{3}$, $S_4$ transmits $X_4(\{3,4\})=C_{d_3,\{4\}}^{4}$, and $S_5$ transmits $X_5(\{3,4\})=C_{d_4,\{3\}}^{5}$. The total number 