On Families of Sets of Integral Vectors Whose Representatives form Sum-Distinct Sets

We study the size of a collection $\{{\cal P}_1,\ldots,{\cal P}_T\}$ whose members ${\cal P}_i$, $i=1,\ldots,T$, are disjoint sets of integral vectors such that $\sum_{i=1}^T \bar x_i$ are all distinct and each $n$-tuple $\bar x_i$ comes from a different set ${\cal P}_i$. In particular, if ${\cal P}_i=\{\bar 0_n,\bar x_i\}$, we have a well-known problem on maximum cardinality of sum-distinct sets of integral vectors. We state bounds on $n^{-1}\sum_{i=1}^T \log_2\vert{\cal P}_i\vert$ and give a construction that meets the lower bound.