For $n\in \mathbb{Z}$ and $A\subseteq \mathbb{Z}$, define $r_{A}(n)$ and ${\it\delta}_{A}(n)$ by $r_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}+a_{2},a_{1}\leq a_{2}\}$ and ${\it\delta}_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}-a_{2}\}$. We call $A$ a unique representation bi-basis if $r_{A}(n)=1$ for all $n\in \mathbb{Z}$ and ${\it\delta}_{A}(n)=1$ for all $n\in \mathbb{Z}\setminus \{0\}$. In this paper, we prove that there exists a unique representation bi-basis $A$ such that $\limsup _{x\rightarrow \infty }A(-x,x)/\sqrt{x}\geq 1/\sqrt{2}$.