{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T09:57:12Z","timestamp":1753437432969,"version":"3.40.5"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T00:00:00Z","timestamp":1686268800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that for a fixed <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline1.png\"\/><jats:tex-math>\n$q$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, the number of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline2.png\"\/><jats:tex-math>\n$q$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-ary <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline3.png\"\/><jats:tex-math>\n$t$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-error correcting codes of length <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline4.png\"\/><jats:tex-math>\n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is at most <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline5.png\"\/><jats:tex-math>\n$2^{(1 + o(1)) H_q(n,t)}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for all <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline6.png\"\/><jats:tex-math>\n$t \\leq (1 - q^{-1})n - 2\\sqrt{n \\log n}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline7.png\"\/><jats:tex-math>\n$H_q(n, t) = q^n\/ V_q(n,t)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the Hamming bound and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline8.png\"\/><jats:tex-math>\n$V_q(n,t)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the cardinality of the radius <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline9.png\"\/><jats:tex-math>\n$t$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000111_inline10.png\"\/><jats:tex-math>\n$t = o(n^{1\/3} (\\log n)^{-2\/3})$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0963548323000111","type":"journal-article","created":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T07:56:58Z","timestamp":1686297418000},"page":"819-832","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["On the number of error correcting codes"],"prefix":"10.1017","volume":"32","author":[{"given":"Dingding","family":"Dong","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0348-5886","authenticated-orcid":false,"given":"Nitya","family":"Mani","sequence":"additional","affiliation":[]},{"given":"Yufei","family":"Zhao","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,6,9]]},"reference":[{"key":"S0963548323000111_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2015.02.005"},{"key":"S0963548323000111_ref3","first-page":"vi+97","article-title":"An algebraic approach to the association schemes of coding theory","volume":"10","author":"Delsarte","year":"1973","journal-title":"Philips Res. Rep. Suppl."},{"year":"1999","author":"van Lint","key":"S0963548323000111_ref12"},{"key":"S0963548323000111_ref11","first-page":"199","article-title":"A survey of perfect codes","volume":"5","author":"van Lint","year":"1975","journal-title":"Rocky Mountain J. Math."},{"key":"S0963548323000111_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-5060(08)70708-8"},{"key":"S0963548323000111_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(82)90204-7"},{"key":"S0963548323000111_ref8","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1109\/TIT.1977.1055688","article-title":"New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities","volume":"23","author":"Robert","year":"1977","journal-title":"IEEE Trans. Inform. 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IMRN"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548323000111","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T12:49:59Z","timestamp":1691498999000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548323000111\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,9]]},"references-count":12,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["S0963548323000111"],"URL":"https:\/\/doi.org\/10.1017\/s0963548323000111","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2023,6,9]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}