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 \((q - 1)m - 2\).

在本文中，我们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\(\mathbb {F}_{q}\)是阶数\(n = q^m\)和q的有限域。我们考虑长度为\(n = q^m\)和阶数为\((q - 1)m - 2\)的广义 Reed-Muller 码。参数与广义里德-穆勒码的参数相同的码称为类里德-穆勒码。我们提出的构造基于将距离 2 MDS 代码划分为阶数为\((q - 1)m - 2\)的类 Reed-Muller 代码。我们在笛卡尔积\ (\mathbb {F}_{n} \times \mathbb {F}_{q}中构造一组\(q^{q^{cn}}\)非等价 1-完美混合码^{n}\)，其中常数c满足\(c < 1\)、\(n = q^m\)且m是足够大的正整数。我们还证明笛卡尔积\(\mathbb {F}_{n} \times \mathbb {F}_{q}^n\)中的每个 1-完美混合码对应于距离-2 的某个分区MDS 代码转换为\((q - 1)m - 2\)阶的 Reed-Muller 类代码。