An almost complete description of perfect codes in direct products of cycles

Let be a 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+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 rn+n(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 other than the constructed ones.