strictly outperform the BHS scheme, then so can a weakly optimal scheme $W$.

Following the same logic, we can define a weakly optimal collection of helper selection schemes:

\[def:collection-scheme-weakly-optimal\] A class/collection of helper selection schemes $\mathcal{W}=\{W_1,W_2,\cdots\}$ is weakly optimal if for any $(n,k,d)$ value, we can always find one $W\in\mathcal{W}$ such that $W$ is weakly optimal.

From Graph-Based Analysis to Explicit Code Construction
-------------------------------------------------------

This Part I of our work focuses exclusively on the graph-based analysis. As discussed in Section \[subsec:ifg\_existing\], the graph-based analysis only gives a necessary condition (cf. [@dimakis2010network]) while the sufficient condition needs to be proved separately through explicit code construction (cf. [@wu2010existence]). Although the graph-based analysis only gives a necessary condition, in the literature of distributed storage, there is not yet any example in which the min-cut-based characterization is provably not achievable by any finite field, which is an evidence of the power/benefits of graph-based analysis.

To complement the necessary conditions derived by the graph-based min-cut analysis in this Part I, we have proved the following (partial) sufficiency statement in Part II [@part2] of this work.

> For any $(n,k,d)$ value, consider the two helper selection schemes proposed in this work, termed the family and the 