  This paper mainly focuses on cones whose basis is a maximum $h$-scattered
linear set. We start by investigating the intersection sizes of such cones with
the hyperplanes. Then we analyze two constructions of point sets with few
intersection sizes with the hyperplanes. In particular, the second one extends
the construction of translation KM-arcs in projective spaces, having as part at
infinity a cone with basis a maximum $h$-scattered linear set. As an instance
of the second construction we obtain cylinders with a hyperoval as basis, which
we call hypercylinders, for which we are able to provide a stability result.
The main motivation for these problems is related to the connections with both
Hamming and rank distance codes. Indeed, we are able to construct codes with
few weights and to provide a stability result for the codes associated with
hypercylinders.
