$k$ value versus repair-bandwidth $\gamma$ curve comparison at the MBR point for $(n,d)=(60,10)$ and file size $\mathcal{M}=1$.[]{data-label="fig:k_vs_gamma_(60-10)"}](k_vs_gamma_60-10){width="47.50000%"}

*Result 6:* Although the main focus of this work is to investigate the benefits of helper selection, a byproduct of our results is a new explicit construction of exact-repair codes for arbitrary $(n,k,d,\alpha,\beta)$ values satisfying $\alpha=d\beta$. This code construction is presented in Part II of this work. Numerically, the proposed codes demonstrate good performance in all $(n,k,d)$ cases. Analytically, it achieves the absolutely optimal MBR points, among all DHS schemes, for all $(n,k,d,\alpha,\beta)$ values satisfying (i) $n\neq 5$, $k=n-1$, and $d=2$; (ii) $n$ is even, $k=n-1$, and $d=3$; (iii) $n\notin \{7,9\}$, $k=n-1$, and $d=4$; (iv) $n$ is even, $n\notin \{8,14\}$, $k=n-1$, and $d=5$; and (v) $n\notin \{10,11,13\}$, $k=n-1$, and $d=6$. This result is the combination of Proposition \[prop:family-plus\_optimal\] and the explicit code construction in Part II.

The Main Results {#sec:results}
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The main result in this paper is the answer to the question “When is it beneficial to choose the good helpers?”. This is stated as a necessary and sufficient condition in the following proposition.

\[prop:comparison\] (*The converse:*) If at least one of the following two conditions is true: (i) $d=1$, $k=3$, and 