On Minimal Asymptotic Bases

Let $\mathbb N$ denote the set of all nonnegative integers, and let $A\subseteq\mathbb N$. Let $h,n\in\mathbb N$, $h\ge 2$ and $r_h(A,n)=\#\{(a_1,\dots,a_h)\in A^h:a_1+\dotsb+a_h=n\}$. The set $A$ is called an asymptotic basis of order $h$ if $r_h(A,n)\ge 1$ for all sufficiently large integer $n$. An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$. Recently, Sun used 2-adic representations of integers to construct a new class of minimal asymptotic bases of order $h$. In this paper, we generalize the 2-adic result to the $g$-adic case.