Perfect codes in direct products of cycles

Let G = × i = 1 n C ℓ i 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 ℓ i is a multiple of r n + ( r + 1 ) n . On the other hand, if a code of G contains a given vertex and its canonical local vertices, then any ℓ i is a multiple of r n + ( r + 1 ) n . 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.