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

Abstract

Let $G={}_{i=1}^n{C}_{{\ell }_i}$ 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 ${\ell }_i$ 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 ${\ell }_i$ 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 other than the constructed ones.