  In this article we construct new minimal intersection sets in
${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we
then use these sets to obtain linear error correcting codes with few weights,
whose weight enumerator we also determine. Furthermore, we provide a new family
of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we
also compute their weight distribution.
