Additive and subtractive bases of $ \mathbb{Z}_m$ in average

Given a positive integer $m$, let $\mathbb{Z}_m$ be the set of residue classes mod $m$. For $A\subseteq \mathbb{Z}_m$ and $n\in \mathbb{Z}_m$, let $\sigma_A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal{H}_m$ be the set of subsets $A\subseteq \mathbb{Z}_m$ such that $\sigma_A(n)\geq1$ for all $n\in \mathbb{Z}_m$. Let $$\ell_m=\min\limits_{A\in \mathcal{H}_m}\left\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\right\rbrace.$$ Following a prior result of Ding and Zhao on Ruzsa's number, we know that $$\limsup_{m\rightarrow\infty}\ell_m\le 192.$$ Ding and Zhao then asked possible improvements on this value. In this paper, we prove $$\limsup\limits_{m\rightarrow\infty}\ell_m\leq 144.$$ Moreover, parallel results on subtractive bases of $\mathbb{Z}_m$ were also investigated here.