Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in $\mathrm{A}\mathrm{G}(r,q^2)$ 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 $\mathrm{P}\mathrm{G}(r,q^2)$ with $r$ even and we also compute their weight distribution.