  Quantum channels, a subset of quantum maps, describe the unitary and
non-unitary evolution of quantum systems. We study a generalization of the
concept of Pauli maps to the case of multipartite high dimensional quantum
systems through the use of the Weyl operators. The condition for such maps to
be valid quantum channels, i.e. complete positivity, is derived in terms of
Fourier transform matrices. From these conditions, we find the extreme points
of this set of channels and identify an elegant algebraic structure nested
within them. In turn, this allows us to expand upon the concept of "component
erasing channels" introduced in earlier work by the authors. We show that these
channels are completely characterized by elements drawn of finite cyclic
groups. An algorithmic construction for such channels is presented and the
smallest subsets of erasing channels which generate the whole set are
determined.
