A subset A of an abelian group G is a B h [ 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 B h [ g ] sets on the abelian groups ( F h , +) ( Z d , +), and ( Z m 1 × … × Z m d , +), for d ≥ 2, h ≥ 2, and g ≥ 1.