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