Perfect mixed codes from generalized Reed–Muller codes

• Published in: Designs, codes, and cryptography codes, and cryptography, 2024-06, Vol.92 (6), p.1747-1759
• Main Author: Romanov, Alexander M.
• Format: Article
• Language: English
• Subjects: Cartesian coordinates; Coding and Information Theory; Computer Science; Cryptology; Discrete Mathematics in Computer Science; Fields (mathematics); Parameters
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-024-01364-3

In this paper, we propose a new method for constructing 1-perfect mixed codes in the Cartesian product F n × F q n , where F n and 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 F n × 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 F n × 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 .