On additive bases of finite abelian groups

Let $G$ be a finite abelian group, let $p$ be the smallest prime divisor of $|G|$, and let $c_0(G)$ be the least integer $t$ such that every regular sequence over $G$ of length at least $t$ is an additive basis. Additive bases have been extensively studied in additive combinatorics, and the exact value of $c_0(G)$ has been determined for many groups of rank at least three and some groups of rank two (in which case groups of the form $C_p\oplus C_{pn}$ turn out to be exceptional). In this paper, we continue the study of $c_0(G)$ and obtain the following results. (1) Together with known results, we prove that $c_0(G)=|G|/p+p-2$ for all abelian groups $G$ with $\text{rank}(G)\geq 3$. (2) For distinct odd primes $p