give a brief preview of our results through concrete examples to illustrate the main contributions of this work. Although we only present here specific examples as a preview, the main results in Section \[sec:results\] are for general $(n,k,d)$ values.

*Result 1:* For $(n,k,d)=(6,3,4)$, RCs with BHS are absolutely optimal, i.e., there exists no RCs with DHS that can outperform BHS.

*Result 2:* For $(n,k,d)=(6,4,4)$, the RCs with the new family helper selection (FHS) scheme proposed in this paper are absolutely optimal in terms of the storage-bandwidth tradeoff among all RCs with DHS, also see Definition \[def:scheme-optimal\]. In Fig. \[fig:storage\_vs\_bandwidth\_(6-4-4)\], the storage-bandwidth tradeoff curve of the FHS scheme, the optimal helper selection scheme, is plotted against the BHS scheme with file size $\mathcal{M}=1$. In Part II, we provide an explicit construction of an exact-repair code that can achieve $(\alpha,\gamma)=(\frac{4}{11},\frac{4}{11})$, the MBR point of the storage-bandwidth tradeoff curve of the FHS scheme in Fig. \[fig:storage\_vs\_bandwidth\_(6-4-4)\]. If we take a closer look at Fig. \[fig:storage\_vs\_bandwidth\_(6-4-4)\], there are 3 corner points on the FHS scheme curve and they are $(\alpha,\gamma)=(0.25,1)$, $(\frac{2}{7},\frac{4}{7})$, and $(\frac{4}{11},\frac{4}{11})$. Since the two corners $(\alpha,\gamma)=(0.25,1)$ and $(\frac{2}{7},\frac{4}{7})$ can be achieved by the scheme in [@wu2010existence] and the new corner point $(\alpha,\gamma)=(\frac{4}{11},\frac{4}{11})$ 