Perfect codes in direct products of cycles

Let G = × i=1 Cli be the direct product of cycles. It is proved that for any r ≥ 1, and any n ≥ 2, each connected component of G contains an r-perfect code provided that each li is a multiple of r n +(r+1). On the other hand, if a code of G contains a given vertex and its canonical local vertices, then any li is a multiple of r + (r + 1). It is also proved that an r-perfect code (r ≥ 2) of G is uniquely determined by n vertices and it is conjectured that for r ≥ 2 no other codes in G exist than the constructed ones.