A subset $\mathcal{A}$ of a finite abelian group $(G,+)$ is called a $B_h$ set on $G$ if all sums of $h$ elements of $\mathcal{A}$ are different . In this paper we state closed formulas for the cardinality of some sets associated with $B_3$ and $B_4$ sets, and we analyze implications for the largest cardinality of a $B_h$ set on $G$ .