{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T17:42:56Z","timestamp":1767634976132,"version":"3.48.0"},"reference-count":14,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T00:00:00Z","timestamp":1767139200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Anhui Province Higher School","award":["2023AH051697"],"award-info":[{"award-number":["2023AH051697"]}]},{"name":"Natural Science Foundation of Anhui Province Higher School","award":["2023AH051682"],"award-info":[{"award-number":["2023AH051682"]}]},{"name":"Natural Science Foundation of Anhui Province Higher School","award":["2022AH020099"],"award-info":[{"award-number":["2022AH020099"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we investigate the m-spotty weight enumerators over the mixed alphabet ZpRk. Specifically, we construct the Gray map from Zp\u03b1\u00d7Rk\u03b2 to Zp\u03b1+k\u03b2, where Rk=Zp+vZp+v2Zp+\u22ef+vk\u22121Zp with vk=0 and k\u22655. Based on this framework, we establish the MacWilliams identity for the m-spotty weight enumerators between a linear code and its dual over ZpRk, by employing the generalized Hadamard transform and the canonical additive character of Zp. Finally, an example is presented to illustrate and validate the theoretical results.<\/jats:p>","DOI":"10.3390\/e28010059","type":"journal-article","created":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:34:23Z","timestamp":1767198863000},"page":"59","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The MacWilliams Identity for the m-Spotty Weight Enumerators over ZpRk"],"prefix":"10.3390","volume":"28","author":[{"given":"Juan","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Base Courses, Anhui Sanlian University, Hefei 230601, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"An","family":"Jiang","sequence":"additional","affiliation":[{"name":"College of General Education, Anhui Wenda University of Information Engineering, Hefei 231201, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4078-8301","authenticated-orcid":false,"given":"Patrick","family":"Sol\u00e9","sequence":"additional","affiliation":[{"name":"Lab I2M (CNRS, Centrale Marseille, Aix-Marseille University), 13009 Marseilles, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1002\/j.1538-7305.1963.tb04003.x","article-title":"A Theorem on the Distribution of Weights in a Systematic Code","volume":"42","author":"MacWilliams","year":"1963","journal-title":"Bell Syst. 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Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1016\/j.jfranklin.2012.06.002","article-title":"The MacWilliams identity for m-spotty Rosenbloom-Tsfasman weight enumerator","volume":"351","author":"Siap","year":"2014","journal-title":"J. Frankl. Inst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s10623-012-9725-z","article-title":"On some new m-spotty Lee weight enumerators","volume":"71","author":"Sharma","year":"2014","journal-title":"Des. Codes Cryptogr."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Shi, M., Choie, Y., Sharma, A., and Sol\u00e9, P. (2020). Codes and Modular Fors: A Dictionary, World Scientific.","DOI":"10.1142\/11628"},{"key":"ref_14","first-page":"641","article-title":"An identity between the m-spotty weight enumerators of a linear code and its dual","volume":"36","author":"Siap","year":"2012","journal-title":"Turk. J. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/59\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T13:43:17Z","timestamp":1767620597000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/59"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,31]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010059"],"URL":"https:\/\/doi.org\/10.3390\/e28010059","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,12,31]]}}}