to $(\tau-1)$ [@dimakis2010network]. Consider any given helper selection scheme $A$ which can be either dynamic or stationary. Since there are infinitely many different failure patterns (since we consider $\tau=1$ to $\infty$), there are infinitely many IFGs corresponding to the same given helper selection scheme $A$. We denote the collection of all such IFGs by $\mathcal{G}_A(n,k,d,\alpha,\beta)$. We define $\mathcal{G}(n,k,d,\alpha,\beta)=\bigcup_{\forall A}\mathcal{G}_A(n,k,d,\alpha,\beta)$ as the union over all possible helper selection schemes $A$. We sometimes drop the input argument and use $\mathcal{G}_A$ and $\mathcal{G}$ as shorthands.

Given an IFG $G\in\mathcal{G}$ and a data collector $t\in \operatorname{DC}(G)$, we use $\operatorname{mincut}_G(s,t)$ to denote the [*minimum cut value*]{} [@west2001introduction] separating $s$, the root node (source node) of $G$, and $t$.

The key reason behind representing the repair problem by an IFG is that it casts the problem as a multicast scenario [@dimakis2010network]. This allows for invoking the results of network coding in [@ahlswede2000network], [@ho2006random]. More specifically, for any helper scheme $A$ and given system parameters $(n,k,d,\alpha,\beta)$, the results in [@ahlswede2000network] prove that the following condition is *necessary* for the RC with helper selection scheme $A$ to satisfy the reliability requirement $$\begin{aligned}
\min_{G\in \mathcal{G}_A}\min_{t\in \operatorname{DC}(G)}\operatorname{mincut}_G(s,t)\geq \mathcal{M}. \label{eq:condition}\end{aligned}$$ If we limit our focus to the BHS scheme, then the 