The Parameters of Reed-Muller Projective mΘ Codes
Abstract:
Reed-Muller codes were originally introduced by Muller in 1954, then Irving Reed gave a decoding method the same year. These codes, of lengths a power of 2, were the first family of codes for which it was pos- sible to de code an infinite number of errors. The finite field underlying the Reed-Muller codes is F2 = {0, 1}. By replacing this field with the mΘ finite field F2Z = {0, 1, 12Z, 32Z}, the study of Reed-Muller codes on F2Z becomes the mΘ Reed-Muller codes. The generalized Reed-Muller codess were intro-duced by Kasami, Lin and Peterson a Weldon. They showed that GRM codes are cyclic and thereby determined the minimum distance. The mΘ generalized Reed Muller codes were developed by Pemha and Tsimi in 2022. Projective Reed-Muller codes are first introduced by Lachaud in 1988 and the dimensions and minimum distances of Projective Reed-Muller codes are determined by Sϕrensen in 1991. In this paper, we intend to define and to present a notion of Reed-Muller Projective mΘ Reed Muller codes, in oth- ers words the Projective Reed-Muller codes on the mΘ field FqZ, q prime or prime power. The nature of the number q will determine the type of Projec- tive Reed-Muller codes. The exact parameters of the Reed-Muller Projective mΘ codes are derived and the dual are characterized. It is shown that the Reed-Muller Projective mΘ codes are an extension of Projective Reed-Muller codes such that the set of mΘ invariants C (Reed Muller Projective mΘ codes) of the mΘ set Reed-Muller Projective mΘ codes is Projective Reed-Muller codes. The Reed-Muller Projective mΘ codes are mΘ cyclic and the generator polynomial is characterized.
