Perfect codes in Doob graphs

We study $1$-perfect codes in Doob graphs $D(m,n)$. We show that such codes that are linear over $GR(4^2)$ exist if and only if $n=(4^{g+d}-1)\mathrm{/}3$ and $m=(4^{g+2d}-4^{g+d})\mathrm{/}6$ for some integers $g\ge 0$ and $d>0$. We also prove necessary conditions on $(m,n)$ for $1$-perfect codes that are linear over $Z_4$ (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 $2m+n=(4^t-1)\mathrm{/}3$ and $m\le (4^t-5\cdot 2^{t-1}+1)\mathrm{/}9$, we prove the existence of $1$-perfect codes in $D(m,n)$, without the restriction to admit some group structure. Keywords: perfect codes, Doob graphs, distance regular graphs.