Perfect codes in direct products of cycles

Let G=×i=1nCℓi 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 rn+(r+1)n. 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 rn+(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.