  The Doob graph $D(m,n)$ is the Cartesian product of $m>0$ copies of the
Shrikhande graph and $n$ copies of the complete graph of order $4$. Naturally,
$D(m,n)$ can be represented as a Cayley graph on the additive group $(Z_4^2)^m
\times (Z_2^2)^{n'} \times Z_4^{n''}$, where $n'+n''=n$. A set of vertices of
$D(m,n)$ is called an additive code if it forms a subgroup of this group. We
construct a $3$-parameter class of additive perfect codes in Doob graphs and
show that the known necessary conditions of the existence of additive
$1$-perfect codes in $D(m,n'+n'')$ are sufficient. Additionally, two
quasi-cyclic additive $1$-perfect codes are constructed in $D(155,0+31)$ and
$D(2667,0+127)$.
