On Perfect Coverings of Two-Dimensional Grids

We study perfect multiple coverings in translation invariant graphs with vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which then express two-variate formal power series. Using known results, conclude that has non-trivial periodizer, is, there exists non-zero polynomial whose product the series presenting is two-periodic configuration. If periodizer line factors at most one direction, to be periodic. result, find many setups where of infinite grids are necessarily also some algorithmic questions on finding coverings.