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

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 ( G, S contains at most |H | − 1 terms from H . Let c0(G) be the smallest integer t such that every regular sequence S over G of length |S| ≥ t forms an additive basis of G, i.e., ∑ (S) = G. The invariant c0(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. 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 c0(G) = pn+2p− 3 where G = Cp ⊕ Cpn with n ≥ 3. We confirm the conjecture for the case when p = 3 and n = q (≥ 5) is a prime number.