{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:29:44Z","timestamp":1760149784626,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,17]],"date-time":"2023-09-17T00:00:00Z","timestamp":1694908800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the probability of an undetected error for general q-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important special case of binary Hamming codes. Finally, several examples are given to illustrate the results of this paper.<\/jats:p>","DOI":"10.3390\/e25091349","type":"journal-article","created":{"date-parts":[[2023,9,17]],"date-time":"2023-09-17T06:12:21Z","timestamp":1694931141000},"page":"1349","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bounds on the Probability of Undetected Error for q-Ary Codes"],"prefix":"10.3390","volume":"25","author":[{"given":"Xuan","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Anhui University, Hefei 230601, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huizhou","family":"Liu","sequence":"additional","affiliation":[{"name":"State Grid Anhui Electric Power Co., Ltd., Hefei 230601, 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":"I2M, CNRS, Aix-Marseille Univetsity, Centrale Marseille, 13009 Marseilles, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1499","DOI":"10.1016\/j.dam.2005.06.014","article-title":"A survey on proper codes","volume":"156","author":"Dodunekova","year":"2008","journal-title":"Discret. 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