A subset A of an abelian group G is a Bh[g] set on G if the elements of G can be written in at most g ways as sum of h elements of A. Given any field F, this work presents constructions of Bh[g] sets on the abelian groups (Fh, +) (Zd, +), and (Zm1 × … × Zmd, +), for d ≥ 2, h ≥ 2, and g ≥ 1.