system designer should be aware that the RCs with BHS in [@dimakis2010network] do not take full advantage of the relatively loose required protection level since we have in this scenario $k>d\geq k^*$.

In this work, we focus on the design target $k$ instead of the actual performance parameter $k^*$, since given the same $k$, the actual $k^*$ value may depend on how we implement the codes. For example, when locally repairable codes [@gopalan2012locality] are used, it is possible to design a system with $k=k^*>d$. However, when RCs are used together with BHS, we may have $k>d\geq k^*$. As we will see later, when RCs are used together with some carefully designed helper selection schemes, we may again achieve $k=k^*>d$. For any given $(n,k,d)$ values, the goal of this paper is to compare the best performance of any possible helper selection scheme that can still satisfy the desired $(n,k,d)$ values regardless whether they offer over-protection ($k> k^*$) or not.

[**The parameter tuple $(n,k,d)$ and other notation:**]{} From the above definitions, the $n$, $k$, and $d$ values must satisfy $$\begin{aligned}
2\leq n,\quad 1\leq k\leq n-1,\quad\text{and}\quad 1\leq d\leq n-1.\label{eq:ccw1}\end{aligned}$$ In all the results in this work, we assume [*implicitly*]{} that the $n$, $k$, 