{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T18:02:29Z","timestamp":1746640949351,"version":"3.37.3"},"reference-count":11,"publisher":"Oxford University Press (OUP)","issue":"12","license":[{"start":{"date-parts":[[2021,9,22]],"date-time":"2021-09-22T00:00:00Z","timestamp":1632268800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007219","name":"Natural Science Foundation of Shanghai","doi-asserted-by":"publisher","award":["21ZR1443000"],"award-info":[{"award-number":["21ZR1443000"]}],"id":[{"id":"10.13039\/100007219","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001459","name":"Singapore Ministry of Education","doi-asserted-by":"crossref","award":["RG12\/19"],"award-info":[{"award-number":["RG12\/19"]}],"id":[{"id":"10.13039\/501100001459","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,12,30]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A $k$-query locally decodable code (LDC) $C$ allows one to encode any $n$-symbol message $x$ as a codeword $C(x)$ of $N$ symbols such that each symbol of $x$ can be recovered by looking at $k$ symbols of $C(x)$, even if a constant fraction of $C(x)$ has been corrupted. Currently, the best known LDCs are matching vector codes (MVCs). A modulus $m=p_1^{\\alpha _1}p_2^{\\alpha _2}\\cdots p_r^{\\alpha _r}$ may result in an MVC with $k\\leq 2^r$ and $N=\\exp (\\exp (O((\\log n)^{1-1\/r} (\\log \\log n)^{1\/r})))$. The $m$ is good if it is possible to have $k&amp;lt;2^r$. The good numbers yield more efficient MVCs. Prior to this work, there are only finitely many good numbers. All of them were obtained via computer search and have the form $m=p_1p_2$. In this paper, we study good numbers of the form $m=p_1^{\\alpha _1}p_2^{\\alpha _2}$. We show that if $m=p_1^{\\alpha _1}p_2^{\\alpha _2}$ is good, then any multiple of $m$ of the form $p_1^{\\beta _1}p_2^{\\beta _2}$ must be good as well. Given a good number $m=p_1^{\\alpha _1}p_2^{\\alpha _2}$, we show an explicit method of obtaining smaller good numbers that have the same prime divisors. Our approach yields infinitely many new good numbers.<\/jats:p>","DOI":"10.1093\/comjnl\/bxab121","type":"journal-article","created":{"date-parts":[[2021,8,11]],"date-time":"2021-08-11T19:11:12Z","timestamp":1628709072000},"page":"2991-2997","source":"Crossref","is-referenced-by-count":1,"title":["On the Modulus in Matching Vector Codes"],"prefix":"10.1093","volume":"65","author":[{"given":"Lin","family":"Zhu","sequence":"first","affiliation":[{"name":"School of Information Science and Technology , ShanghaiTech University, Shanghai 201210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wen Ming","family":"Li","sequence":"additional","affiliation":[{"name":"School of Information Science and Technology , ShanghaiTech University, Shanghai 201210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liang Feng","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Information Science and Technology , ShanghaiTech University, Shanghai 201210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2021,9,22]]},"reference":[{"key":"2023010312515199200_ref1","first-page":"80","volume-title":"Proc. 32nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2000","author":"Katz","year":"2000"},{"key":"2023010312515199200_ref2","first-page":"72","article-title":"A survey on private information retrieval","volume":"82","author":"Gasarch","year":"2004","journal-title":"Bulletin of the EATCS"},{"year":"2004","author":"Trevisan","key":"2023010312515199200_ref3"},{"key":"2023010312515199200_ref4","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s00037-006-0216-3","article-title":"Lower bounds for linear locally decodable codes and private information retrieval","volume":"15","author":"Goldreich","year":"2006","journal-title":"Comput. 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