Perfect mixed codes from generalized Reed-Muller codes

Abstract

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 $(q-1)m-2$.