above necessary condition becomes $$\begin{aligned}
\min_{G\in \mathcal{G}}\min_{t\in \operatorname{DC}(G)}\operatorname{mincut}_G(s,t)\geq \mathcal{M}.\label{eq:condition-BR}\end{aligned}$$ An important contribution of [@dimakis2010network] is a closed-form expression of the left-hand side (LHS) of $$\begin{aligned}
\min_{G\in \mathcal{G}}\min_{t\in \operatorname{DC}(G)}\operatorname{mincut}_G(s,t)=\sum_{i=0}^{k-1} \min((d-i)^+\beta,\alpha),\label{eq:ex_low_b}\end{aligned}$$ which allows us to numerically check whether is true.

Reference [@wu2010existence] further proves that is not only necessary but also sufficient for the existence of a blind RC with some finite field $\operatorname{GF}(q)$ that satisfies the reliability requirement. Namely, as long as “the right-hand side (RHS) of $\geq \mathcal{M}$" is true, then there exists a RC that meets the system design parameters $(n,k,d,\alpha,\beta)$ even for the worst possible helper selection scheme (since we take the minimum over $\mathcal{G}$).

In contrast with the existing results on the BHS scheme, this work focuses on any given helper selection scheme $A$ and we are thus interested in the bandwidth-storage tradeoff specified in instead of . For example, the Minimum Bandwidth Regenerating (MBR) and Minimum Storage Regenerating (MSR) points of a given helper selection scheme $A$ can be defined by

For any given $(n,k,d)$ values, the MBR point $(\alpha_{\text{MBR}}, \beta_{\text{MBR}})$ of a helper scheme $A$ is defined by $$\begin{aligned}
\beta_{\text{MBR}}\stackrel{\Delta}{=}\min \{\beta: (\alpha,\beta)\text{ satisfies \eqref{eq:condition} and } \alpha=\infty\}\label{eq:MBR-beta-def}\\
\alpha_{\text{MBR}}\stackrel{\Delta}{=}\min \{\alpha: (\alpha,\beta)\text{ satisfies \eqref{eq:condition} and } \beta=\beta_{\text{MBR}}\}.\label{eq:MBR-alpha-def}\end{aligned}$$

