$t$-Intersection sets in $AG(r,q^2)$ and two-character multisets in $PG(3,q^2)$

In this article we construct new minimal intersection sets in $AG(r,q^2)$ with respect to hyperplanes, of size $q^2r-1$ and multiplicity $t$, where $t\in \ q^2r-3-q^(3r-5)/2, q^2r-3+q^(3r-5)/2-q^(3r-3)/2\$, for $r$ odd or $t \in \ q^2r-3-q^(3r-4)/2, q^2r-3-q^r-2\$, for $r$ even. As a byproduct, for any odd $q$ we get a new family of two-character multisets in $PG(3,q^2)$. The essential idea is to investigate some point-sets in $AG(r,q^2)$ satisfying the opposite of the algebraic conditions required in [1] for quasi--Hermitian varieties.