\(d\)-dimensional two-character \(k\)-sets in an affine space \(\mathrm{AG}(r,q)\) (Q793337)

In \(\mathrm{AG}(r,q)\) with \(r\geq 4\), a \(k\)-set is of type \((m,n)_ d\) with \(0\leq m<n\leq q^ d\) if every \(d\)-dimensional subspace of \(\mathrm{AG}(r,q)\) meets it in either \(m\) or \(n\) points (and such \(m\)-secant and \(n\)-secant subspaces exist). The author characterizes these sets completely for \(d\leq r-2\).