  The Doob scheme $D(m,n'+n'')$ is a metric association scheme defined on
$E_4^m \times F_4^{n'}\times Z_4^{n''}$, where $E_4=GR(4^2)$ or, alternatively,
on $Z_4^{2m} \times Z_2^{2n'} \times Z_4^{n''}$. We prove the MacWilliams
identities connecting the weight distributions of a linear or additive code and
its dual. In particular, for each case, we determine the dual scheme, on the
same set but with different metric, such that the weight distribution of an
additive code $C$ in the Doob scheme $D(m,n'+n'')$ is related by the
MacWilliams identities with the weight distribution of the dual code $C^\perp$
in the dual scheme. We note that in the case of a linear code $C$ in $E_4^m
\times F_4^{n'}$, the weight distributions of $C$ and $C^\perp$ in the same
scheme are also connected.
