Upper bounds for finite additive $2$-bases

For a positive integer $N$, a set $\mathcal{A}\subset [0,N]\cap\mathbb{Z}$ is called a $2$-basis for $N$ if every integer $n\in [0,N]$ can be represented as $n=a+b$, where $a, b\in\mathcal{A}$. In this paper, we give a lower bound estimate for the cardinality of an additive $2$-basis for $N$, as $N\to\infty$, which improves the existing results on this topic.