Optimal data placements for triple replication

Given a set $V$ of $v$ servers along with $b$ files (data), each file is replicated (placed) on exactly $k$ servers and thus a file can be represented by a set of $k$ servers. Then we produce a data placement consisting of $b$ subsets of $V$ called blocks, each of size $k$. Each server has some probability to fail and we want to find a placement that minimizes the variance of the number of available files. It was conjectured that there always exists an optimal data placement (with variance better than any other placement for any value of the probability of failure). An optimal data placement for triple replication with $b$ blocks (of size three) on a $v$-set was proved to exist by Wei et al. if $v$ and $b$ are not excluded by two conditions. This article concentrates on the parameters $v, b$ satisfying the two conditions and characterizes the combinatorial properties of the corresponding optimal data placements. Nearly well-balanced triple systems (NWBTSs) are defined to produce optimal data placements. Many constructions for NWBTSs are developed, mainly by constructing candelabra systems with various desirable partitions. The main result of this article is that there always exist optimal data placements for triple replication with $b$ blocks on a $v$-set possibly except when $v\equiv 4$ (mod 24) or $v=50,74$, and $\frac{\lambda v(v-1)}{6}-\frac{v}{6} < b < \frac{\lambda v(v-1)}{6}+\frac{v}{6}$ for an odd integer $\lambda$.