MDS codes in the Doob graphs

The Doob graph $D(m,n)$, where $m>0$, is the direct product of $m$ copies of The Shrikhande graph and $n$ copies of the complete graph $K_4$ on $4$ vertices. The Doob graph $D(m,n)$ is a distance-regular graph with the same parameters as the Hamming graph $H(2m+n,4)$. In this paper we consider MDS codes in Doob graphs with code distance $d \ge 3$. We prove that if $2m+n>6$ and $2<d<2m+n$, then there are no MDS codes with code distance $d$. We characterize all MDS codes with code distance $d \ge 3$ in Doob graphs $D(m,n)$ when $2m+n \le 6$. We characterize all MDS codes in $D(m,n)$ with code distance $d=2m+n$ for all values of $m$ and $n$.