Work

Generator Matrices and Symmetrized Weight Enumerators of Linear Codes over Fpm + uFpm + vFpm + wFpm

Year: 2024

Type: article

Abstract: Let u,v, and w be indeterminates over Fpm and let R=Fpm+uFpm+vFpm+wFpm, where p is a prime. Then, R is a ring of order p4m, and R≅Fpm[u,v,w]I with maximal ideal J=uFpm+vFpm+wFpm of order p3m and a residue field Fpm of order pm, where I is an appropriate ideal. In this article, the goal is to improve the understanding of linear codes over local non-chain rings. In particular, we investigate the symmetrized weight enumerators and generator matrices of linear codes of length N over R. In order to accomplish that, we first list all such rings up to the isomorphism for different values of the index of nilpotency l of J, 2≤l≤4. Furthermore, we fully describe the lattice of ideals of R and their orders. Next, for linear codes C over R, we compute the generator matrices and symmetrized weight enumerators, as shown by numerical examples.

Source: Symmetry

DOI: 10.3390/sym16091169

Authors: Alhanouf Ali Alhomaidhi, Sami Alabiad, Nawal A. Alsarori

Institutions: King Saud University, Dr. Babasaheb Ambedkar Marathwada University

Language: English

Topic: Coding theory and cryptography

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,728

Funder: King Saud University