Characterizing r-perfect codes in direct products of two and three cycles

An r-perfect code of a graph G=(V,E) is a set C⊆V such that the r-balls centered at vertices of C form a partition of V. It is proved that the direct product of Cm and Cn (r⩾1, m,n⩾2r+1) contains an r-perfect code if and only if m and n are each a multiple of (r+1)2+r2 and that the direct product of Cm, Cn, and Cℓ (r⩾1, m,n,ℓ⩾2r+1) contains an r-perfect code if and only if m, n, and ℓ are each a multiple of r3+(r+1)3. The corresponding r-codes are essentially unique. Also, r-perfect codes in C2r×Cn (r⩾2, n⩾2r) are characterized.