2023/12/26 by Alexander M. Romanov, Romanov, Alexander M.
Computer Science · Engineering · #Cellular Automata and Applications #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2312.15937
openalex publication_date 2023/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we propose a new method for constructing 1-perfect mixed codes in the Cartesian product \mathbbFn × \mathbbFqn, where \mathbbFn and \mathbbFq are finite fields of orders n = qm and q. We consider generalized Reed-Muller codes of length n = qm 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^qcn nonequivalent 1-perfect mixed codes in the Cartesian product \mathbbFn × \mathbbFqn, where the constant c satisfies c < 1, n = qm and m is a sufficiently large positive integer. We also prove that each 1-perfect mixed code in the Cartesian product \mathbbFn × \mathbbFqn corresponds to a certain partition of a distance-2 MDS code into Reed-Muller-like codes of order (q - 1)m - 2.