Additive bases of $C_3\oplus C_{3q}$

Abstract: Let $G$ be a finite abelian group and $p$ be the smallest prime dividing $|G|$. Let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subsetneq G$, $S$ contains at most $|H|-1$ terms from $H$. Let $\mathsf c_0(G)$ be the smallest integer $t$ such that every regular sequence $S$ over $G$ of length $|S|\geq t$ forms an additive basis of $G$, i.e., $\sum(S)=G$. The invariant $\mathsf c_0(G)$ was first studied by Olson and Peng in 1980's, and since then it has been determined for all finite abelian groups except for the groups with rank 2 and a few groups of rank 3 or 4 with order less than $10^8$. In this paper, we focus on the remaining case concerning groups of rank 2. It was conjectured by the first author and Han (Int. J. Number Theory 13 (2017) 2453-2459) that $\mathsf c_0(G)=pn+2p-3$ where $G=C_p\oplus C_{pn}$ with $n\geq 3$. We confirm the conjecture for the case when $p=3$ and $n=q \,(\geq 5)$ is a prime number.