{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:34:00Z","timestamp":1760146440169,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,2]],"date-time":"2024-11-02T00:00:00Z","timestamp":1730505600000},"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":["Symmetry"],"abstract":"<jats:p>Suppose R is a finite commutative local ring, then it is known that R has four positive integers p,n,m,k called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field Fpm and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order p4m. Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants p,n,m,k. Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings.<\/jats:p>","DOI":"10.3390\/sym16111455","type":"journal-article","created":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T03:57:34Z","timestamp":1730692654000},"page":"1455","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Frobenius Local Rings of Order p4m"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-2520-2699","authenticated-orcid":false,"given":"Alhanouf Ali","family":"Alhomaidhi","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\/0000-0002-6824-6985","authenticated-orcid":false,"given":"Sami","family":"Alabiad","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":[[2024,11,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1216\/rmjm\/1181072765","article-title":"A note of finite local rings","volume":"22","author":"Whelan","year":"1992","journal-title":"Rocky Mt. J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Alabiad, S., Alhomaidhi, A.A., and Alsarori, N.A. (2024). On linear codes over local rings of order p4. Mathematics, 12.","DOI":"10.3390\/math12193069"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1007\/PL00000451","article-title":"Characterization of finite Frobenius rings","volume":"76","author":"Honold","year":"2001","journal-title":"Arch. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1353\/ajm.1999.0024","article-title":"Duality for modules over finite rings and applications to coding theory","volume":"121","author":"Wood","year":"1999","journal-title":"Am. J. Math."},{"key":"ref_5","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. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Alhomaidhi, A.A., Alabiad, S., and Alsarori, N.A. (2024). Generator matrices and symmetrized weight enumerators of linear codes over Fpm+uFpm+vFpm+wFpm. Symmetry, 16.","DOI":"10.3390\/sym16091169"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Alabiad, S., Alhomaidhi, A.A., and Alsarori, N.A. (2024). On linear codes over finite singleton local rings. Mathematics, 12.","DOI":"10.3390\/math12071099"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ffa.2016.08.004","article-title":"Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3","volume":"43","year":"2017","journal-title":"Finite Fields Their Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1006\/jabr.2000.8350","article-title":"Rings of order p5 Part II. Local Rings","volume":"231","author":"Corbas","year":"2000","journal-title":"J. Algebra"},{"key":"ref_10","unstructured":"McDonald, B.R. (1974). Finite Rings with Identity, Marcel Dekker."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zariski, O., and Samuel, P. (1960). Commutative Algebra, Springer.","DOI":"10.1007\/978-3-662-29244-0"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Matsumura, H. (1986). Commutative Ring Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781139171762"},{"key":"ref_13","first-page":"195","article-title":"Finite associative rings","volume":"21","author":"Raghavendran","year":"1969","journal-title":"Compos. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"643","DOI":"10.2140\/pjm.1974.53.643","article-title":"Representations of finite rings","volume":"53","author":"Wilson","year":"1974","journal-title":"Pac. J. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0022-314X(72)90070-4","article-title":"On the group of units of certain rings","volume":"4","author":"Ayoub","year":"1972","journal-title":"J. Number Theory"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1455\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:27:17Z","timestamp":1760113637000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1455"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,2]]},"references-count":15,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["sym16111455"],"URL":"https:\/\/doi.org\/10.3390\/sym16111455","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,11,2]]}}}