while using strictly less than the bandwidth of the MBR point of the original RC ($(\alpha,\gamma)=(\frac{2}{3},\frac{2}{3})$). Since the provably optimal FHS scheme has only a single corner point, it means that we can achieve minimum-storage (the MSR point) and minimum-bandwidth (the MBR point) simultaneously.

![Storage-bandwidth tradeoff curves of RCs with BHS versus RCs with the absolutely optimal scheme (FHS) for $(n,k,d)=(5,3,2)$ and file size $\mathcal{M}=1$.[]{data-label="fig:storage_vs_bandwidth_(5-3-2)"}](storage_vs_bandwidth_5-3-2){width="47.50000%"}

*Result 4:* For $(n,k,d)=(20,10,10)$, we do not know what is the absolutely optimal DHS scheme. On the other hand, the FHS scheme again outperforms the BHS scheme. Fig. \[fig:storage\_vs\_bandwidth\_(20-10-10)\] shows a tradeoff curve comparison between the FHS scheme and the BHS scheme.

![Storage-bandwidth tradeoff curves of RCs with BHS versus RCs with FHS for $(n,k,d)=(20,10,10)$ and file size $\mathcal{M}=1$.[]{data-label="fig:storage_vs_bandwidth_(20-10-10)"}](storage_vs_bandwidth_20-10-10){width="47.50000%"}

*Result 5:* For $(n,d)=(60,10)$, we do not know what is the absolutely optimal DHS scheme. However, in Fig. \[fig:k\_vs\_gamma\_(60-10)\], we plot a $k$ versus repair-bandwidth curve to compare the BHS scheme to the FHS scheme while restricting to the MBR point. Examining Fig. \[fig:k\_vs\_gamma\_(60-10)\], we can see that the BHS scheme performs poorly compared to FHS as $k$ grows larger. When $k=d=10$, the FHS scheme only uses $73.33\%$ of the bandwidth of the BHS scheme.

![The 