Perfect codes in Doob graphs

We study 1-perfect codes in Doob graphsD(m,n). We show that such codes that are linear over GR(4) exist if and only if n = (4γ+δ−1)/3 andm = (4γ+2δ−4γ+δ)/6 for some integers γ ≥ 0 and δ > 0. We also prove necessary conditions on (m,n) for 1-perfect codes that are linear over Z4 (we call such codes additive) to exist in D(m,n) graphs; for some of these parameters, we show the existence of codes. For every m and n satisfying 2m + n = (4μ − 1)/3 and m ≤ (4μ − 5 · 2μ−1 + 1)/9, we prove the existence of 1-perfect codes in D(m,n), without the restriction to admit some group structure.