On Linear Codes over Local Rings of Order p 4

Sami Alabiad (), Alhanouf Ali Alhomaidhi and Nawal A. Alsarori
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Sami Alabiad: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Alhanouf Ali Alhomaidhi: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Nawal A. Alsarori: Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India

Mathematics, 2024, vol. 12, issue 19, 1-20

Abstract: Suppose R is a local ring with invariants p , n , r , m , k and m r = 4 , that is R of order p 4 . Then, R = R 0 + u R 0 + v R 0 + w R 0 has maximal ideal J = u R 0 + v R 0 + w R 0 of order p ( m − 1 ) r and a residue field F of order p r , where R 0 = G R ( p n , r ) is the coefficient subring of R . In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over R . In order to accomplish that, we first list all such rings up to isomorphism for different values of p , n , r , m , k . Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R , we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them.

Keywords: MacWiliams identities; coding over rings; local rings; generator matrices (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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