We show there is an uncountable number of parallel total perfect codes in the
integer lattice graph Ξ of R2. In contrast, there is just one
1-perfect code in Ξ and one total perfect code in Ξ
restricting to total perfect codes of rectangular grid graphs (yielding an
asymmetric, Penrose, tiling of the plane). We characterize all cycle products
CmβΓCnβ with parallel total perfect codes, and the d-perfect and
total perfect code partitions of Ξ and CmβΓCnβ, the former
having as quotient graph the undirected Cayley graphs of Z2d2+2d+1β with
generator set {1,2d2}. For r>1, generalization for 1-perfect codes is
provided in the integer lattice of Rr and in the products of r cycles,
with partition quotient graph K2r+1β taken as the undirected Cayley graph
of Z2r+1β with generator set {1,...,r}.Comment: 16 pages; 11 figures; accepted for publication in Austral. J. Combi