Perfect domination in regular grid graphs

We show there is an uncountable number of parallel total perfect codes in the integer lattice graph Λ of R. In contrast, there is just one 1-perfect code in Λ and one total perfect code in Λ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products Cm × Cn with parallel total perfect codes, and the d-perfect and total perfect code partitions of Λ and Cm×Cn, the former having as quotient graph the undirected Cayley graphs of Z2d2+2d+1 with generator set {1, 2d }. For r > 1, generalization for 1-perfect codes is provided in the integer lattice of R and in the products of r cycles, with partition quotient graph K2r+1 taken as the undirected Cayley graph of Z2r+1 with generator set {1, . . . , r}.