Projective Two-Weight Sets of Denniston Type

We construct two-weight sets in PG$(3n-1,q)$, $n\geq2$ with the same weights as those that would arise from the blow-up of a maximal $q$-arc in PG$(2,q^n)$. The construction is of particular interest when $q$ is odd, as it is well known that no maximal arcs in PG$(2,q^n)$ exist in that case.