On $B_h[1]$-sets which are asymptotic bases of order $2h$

Let $h,k\ge 2$ be integers. A set $A$ of positive integers is called asymptotic basis of order $k$ if every large enough positive integer can be written as the sum of $k$ terms from $A$. A set of positive integers $A$ is said to be a $B_h[g]$-set if every positive integer can be written as the sum of $h$ terms from $A$ at most $g$ different ways. In this paper we prove the existence of $B_h[1]$ sets which are asymptotic bases of order $2h$ by using probabilistic methods.