  Discretizations of continuum theories often do not preserve the gauge
symmetry content. This occurs in particular for diffeomorphism symmetry in
general relativity, which leads to severe difficulties both in canonical and
covariant quantization approaches. We discuss here the method of perfect
actions, which attempts to restore gauge symmetries by mirroring exactly
continuum physics on a lattice via a coarse graining process. Analytical
results can only be obtained via a perturbative approach, for which we consider
the first steps, namely the coarse graining of the linearized theory. The
linearized gauge symmetries are exact also in the discretized theory, hence we
develop a formalism to deal with gauge systems. Finally we provide a
discretization of linearized gravity as well as a coarse graining map and show
that with this choice the 3D linearized gravity action is invariant under
coarse graining.
