{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T10:49:46Z","timestamp":1761130186584,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,15]],"date-time":"2025-03-15T00:00:00Z","timestamp":1741996800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSPD2025R871"],"award-info":[{"award-number":["RSPD2025R871"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let R=Fpm[u,v] be the ring whose order is p4m, where p denotes a prime number. The algebraic structure of R is substantially influenced by the relationships among the elements u2, uv, and v2. We establish that R is formed over Fpm when p\u22602 and u2=\u03b3v2, where \u03b3 is a non-zero element of Fpm and uv=0. For p=2, the situations are those where u2=v2=0 and uv\u22600. After that, we discuss how to make matrices that are linked to the symmetrical weight enumerators of linear codes over these rings. These matrices are essential in the examination of coding theory over Frobenius rings. Utilizing these matrices makes it possible to determine the homogeneous distances of linear codes over the ring R. Our results show a connection between the uniform distances of linear codes and the matrices of symmetrized weight enumerators. This helps us understand the behavior and characteristics of these codes.<\/jats:p>","DOI":"10.3390\/sym17030440","type":"journal-article","created":{"date-parts":[[2025,3,17]],"date-time":"2025-03-17T08:57:02Z","timestamp":1742201822000},"page":"440","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Homogeneous Weights of Linear Codes over \ud835\udd3dpm[u, v] of Order p4m"],"prefix":"10.3390","volume":"17","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"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,15]]},"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. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/S1071-5797(03)00007-8","article-title":"Homogeneous weights and exponential sums","volume":"9","author":"Voloch","year":"2003","journal-title":"Finite Fields Their Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"101892","DOI":"10.1016\/j.ffa.2021.101892","article-title":"LCD codes from tridiagonal Toeplitz matrices","volume":"75","author":"Shi","year":"2021","journal-title":"Finite Fields Their Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1016\/j.jfranklin.2012.05.014","article-title":"A class of optimal p-ary codes from one-weight codes over Fp[u]\u2329um\u232a","volume":"350","author":"Shi","year":"2013","journal-title":"J. Frankl. Inst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1060","DOI":"10.1109\/18.841186","article-title":"On the Hamming distance of linear codes over a finite chain ring","volume":"46","author":"Norton","year":"2000","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_6","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_7","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_8","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_9","unstructured":"McDonald, B.R. (1974). Finite Rings with Identity, Marcel Dekker."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Zariski, O., and Samuel, P. (1960). Commutative Algebra, Springer.","DOI":"10.1007\/978-3-662-29244-0"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Matsumura, H. (1986). Commutative Ring Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781139171762"},{"key":"ref_12","first-page":"195","article-title":"Finite associative rings","volume":"21","author":"Raghavendran","year":"1969","journal-title":"Compos. Math."},{"key":"ref_13","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_14","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"},{"key":"ref_15","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_16","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_17","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 \ud835\udd3dpm + u\ud835\udd3dpm + v\ud835\udd3dpm + w\ud835\udd3dpm. Symmetry, 16.","DOI":"10.3390\/sym16091169"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/440\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:54:15Z","timestamp":1760028855000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/440"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,15]]},"references-count":17,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["sym17030440"],"URL":"https:\/\/doi.org\/10.3390\/sym17030440","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,3,15]]}}}