is proved to be achievable in Part II, we can thus achieve the entire optimal tradeoff curve in Fig. \[fig:storage\_vs\_bandwidth\_(6-4-4)\] by space-sharing while no other scheme can do better, as stated in Proposition \[prop:optimal\_main\].[^4]

![Storage-bandwidth tradeoff curves of RCs with BHS versus RCs with the absolutely optimal scheme (FHS) for $(n,k,d)=(6,4,4)$ and file size $\mathcal{M}=1$.[]{data-label="fig:storage_vs_bandwidth_(6-4-4)"}](storage_vs_bandwidth_6-4-4){width="47.50000%"}

*Result 3:* For $(n,k,d)=(5,3,2)$, the proposed FHS scheme again outperforms the BHS scheme, and is provably optimal.[^5] We note that BHS is inherently inefficient in this example since BHS always has $k^*\leq d$ and thus overprotects the data when $d<k$. However, for this particular $(n,k,d)$ combination we do not have any other existing scheme that can be used as a baseline. For that reason, we still compare to BHS in this example for the sake of illustration. Fig. \[fig:storage\_vs\_bandwidth\_(5-3-2)\] shows a tradeoff curve comparison between the FHS scheme and the BHS scheme. An interesting phenomenon is that the tradeoff curve of the FHS scheme has only one corner point $(\alpha,\gamma)=(0.5,0.5)$ and we can achieve this point by an exact-repair scheme, see Part II [@part2]. Note that this exact-repair scheme for $(\alpha,\gamma)=(0.5,0.5)$ has the same storage consumption as the MSR point of the original RC ($(\alpha,\gamma)=(0.5,1)$) 