Minimal asymptotic $g$-adic bases from arbitrary partitions

Let $g>h\ge 2$ be integers. For a nonempty set $W\subseteq \mathbb{N}$, let ${\mathcal{F}}^{\ast }(W)$ denote the set of its finite nonempty subsets. Let $$A_g(W)=\{\sum _{f\in F}{a}_f{g}^f:1\le a_f\le g-1,F\in {\mathcal{F}}^{\ast }(W)\},$$ and let $\mathbb{N}=W_1\bigsqcup \cdots \bigsqcup W_h$ be a partition into infinite sets. We prove that $A={\bigcup }_{i=1}^h{A}_g(W_i)$ is a minimal asymptotic basis of order $h$ for every such partition.