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Perfect mixed codes from generalized Reed–Muller codes

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

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  1. Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090

    Alexander M. Romanov

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Communicated by V. A. Zinoviev.

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Romanov, A.M. Perfect mixed codes from generalized Reed–Muller codes. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01364-3

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