On perfect codes in Cartesian products of graphs

Assuming the existence of a partition in perfect codes of the vertex set of a finite or infinite bipartite graph GG we give the construction of a perfect code in the Cartesian product G□G□P2G□G□P2. Such a partition is easily obtained in the case of perfect codes in Abelian Cayley graphs and we give some examples of applications of this result and its generalizations.