On multifold perfect codes and some other completely regular codes in the Doob graphs and quaternary Hamming graphs

Abstract

We consider the problem of existence of perfect $2$-colorings in the Doob graphs $D(m,n)$ and $4$-ary Hamming graphs $H(n,4)$. We characterize all parameters for which multifold $1$-perfect code in $D(m,n)$ exists. Also, we prove that for any pair $(b,c)$ that satisfy standard conditions (Lloyd's and sphere-packing conditions) there is perfect $(b,c)$-coloring in Doob graphs and Hamming graphs if diameter of graph and $n$ are sufficiently large ($n$ not less than $0$, $1$ or $8$ for some cases). Also we obtain some completely regular codes with covering radius $2$.