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 $(q - 1)m - 2$.

翻译：本文提出了一种在笛卡尔积 $\mathbb{F}_{n} \times \mathbb{F}_{q}^n$ 中构造 $1$-完美混合码的新方法，其中 $\mathbb{F}_{n}$ 和 $\mathbb{F}_{q}$ 分别为阶数 $n = q^m$ 和 $q$ 的有限域。我们考虑长度为 $n = q^m$、阶数为 $(q - 1)m - 2$ 的广义Reed-Muller码。参数与广义Reed-Muller码相同的码称为类Reed-Muller码。所提出的构造基于将距离为2的MDS码划分为阶数为 $(q - 1)m - 2$ 的类Reed-Muller码。我们在笛卡尔积 $\mathbb{F}_{n} \times \mathbb{F}_{q}^{n}$ 中构造了 $q^{q^{cn}}$ 个非等价的 $1$-完美混合码，其中常数 $c$ 满足 $c < 1$，$n = q^m$，且 $m$ 为足够大的正整数。我们还证明了笛卡尔积 $\mathbb{F}_{n} \times \mathbb{F}_{q}^n$ 中的每个 $1$-完美混合码对应于将距离为2的MDS码划分为阶数为 $(q - 1)m - 2$ 的类Reed-Muller码的特定划分。