{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T20:18:41Z","timestamp":1772050721773,"version":"3.50.1"},"reference-count":16,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T00:00:00Z","timestamp":1723593600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSPD2024R871"],"award-info":[{"award-number":["RSPD2024R871"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Suppose that R=Zp4[u] with u2=p3\u03b2 and pu=0, where p is a prime and \u03b2 is a unit in R. Then, R is a local non-chain ring of order p5 with a unique maximal ideal J=(p,u) and a residue field of order p. A linear code C of length N over R is an R-submodule of RN. The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over R. We first prove that when p\u22602, there are precisely two distinct rings with these properties up to isomorphism. However, for p=2, only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over R, illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings.<\/jats:p>","DOI":"10.3390\/axioms13080552","type":"journal-article","created":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T03:46:36Z","timestamp":1723607196000},"page":"552","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["MacWilliams Identities and Generator Matrices for Linear Codes over \u2124p4[u]\/(u2 \u2212 p3\u03b2, pu)"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6824-6985","authenticated-orcid":false,"given":"Sami","family":"Alabiad","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-2520-2699","authenticated-orcid":false,"given":"Alhanouf Ali","family":"Alhomaidhi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9691-4979","authenticated-orcid":false,"given":"Nawal A.","family":"Alsarori","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/PL00012382","article-title":"On the structure of linear cyclic codes over finite chain rings","volume":"10","author":"Norton","year":"2000","journal-title":"Appl. Algebra Eng. Commun. 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