  In this paper, we propose a new method for constructing $1$-perfect mixed
codes in the Cartesian product $\mathbb{F}_{n} \times \mathbb{F}_{q}^n$, where
$\mathbb{F}_{n}$ and $\mathbb{F}_{q}$ are finite fields of orders $n = q^m$ and
$q$. We consider generalized Reed-Muller codes of length $n = q^m$ and order
$(q - 1)m - 2$. Codes whose parameters are the same as the parameters of
generalized Reed-Muller codes are called Reed-Muller-like codes. The
construction we propose is based on partitions of distance-2 MDS codes into
Reed-Muller-like codes of order $(q - 1)m - 2$. We construct a set of
$q^{q^{cn}}$ nonequivalent 1-perfect mixed codes in the Cartesian product
$\mathbb{F}_{n} \times \mathbb{F}_{q}^{n}$, where the constant $c$ satisfies $c
< 1$, $n = q^m$ and $m$ is a sufficiently large positive integer. We also prove
that each $1$-perfect mixed code in the Cartesian product $\mathbb{F}_{n}
\times \mathbb{F}_{q}^n$ corresponds to a certain partition of a distance-2 MDS
code into Reed-Muller-like codes of order $(q - 1)m - 2$.
