  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 $n$.
