An almost complete description of perfect codes in direct products of cyclesKlavžar, Sandi (Avtor) Špacapan, Simon (Avtor) Žerovnik, Janez (Avtor) mathematicsgraph theoryerror-correcting codesdirect product of graphsperfect codescyclesLet ▫$G = times_{i=1}^nC_{ell_i}$▫ be a direct product of cycles. It is proved that for any ▫$r ge 1$▫, and any ▫$n ge 2$▫, 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 ▫$(r ge 2)$▫ of ▫$G$▫ is uniquely determined by ▫$n$▫ vertices, and it is conjectured that for ▫$r ge 2$▫ no other codes in ▫$G$▫ exist other than the constructed ones.Elsevier20062015-07-10 14:54:37Znanstveno delo51554sl