  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.
