Perfect mixed codes from generalized Reed–Muller codes

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