{"title":"由车域覆盖","authors":"Eugene R. Rodemich","doi":"10.1016/S0021-9800(70)80018-7","DOIUrl":null,"url":null,"abstract":"<div><p>It is shown that a 1-dense set in <em>V<sub>n</sub><sup>k</sup></em> must contain at least <em>n<sup>k−1</sup>/(k−1)</em> points. As a corollary, a conjecture of Golomb and Posner on error-distributing codes is proved. It is also shown that a (<em>k</em>−2)-dense set must contain at least <em>n</em><sup>2</sup>/(<em>k</em>−1) points. Equality can be attained if and only if <em>k</em>−1 divides <em>n</em> and there are <em>k</em>−2 orthogonal latin squares of order <em>n</em>/(<em>k</em>−1).</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 2","pages":"Pages 117-128"},"PeriodicalIF":0.0000,"publicationDate":"1970-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80018-7","citationCount":"23","resultStr":"{\"title\":\"Coverings by rook domains\",\"authors\":\"Eugene R. Rodemich\",\"doi\":\"10.1016/S0021-9800(70)80018-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>It is shown that a 1-dense set in <em>V<sub>n</sub><sup>k</sup></em> must contain at least <em>n<sup>k−1</sup>/(k−1)</em> points. As a corollary, a conjecture of Golomb and Posner on error-distributing codes is proved. It is also shown that a (<em>k</em>−2)-dense set must contain at least <em>n</em><sup>2</sup>/(<em>k</em>−1) points. Equality can be attained if and only if <em>k</em>−1 divides <em>n</em> and there are <em>k</em>−2 orthogonal latin squares of order <em>n</em>/(<em>k</em>−1).</p></div>\",\"PeriodicalId\":100765,\"journal\":{\"name\":\"Journal of Combinatorial Theory\",\"volume\":\"9 2\",\"pages\":\"Pages 117-128\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1970-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80018-7\",\"citationCount\":\"23\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021980070800187\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021980070800187","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
It is shown that a 1-dense set in Vnk must contain at least nk−1/(k−1) points. As a corollary, a conjecture of Golomb and Posner on error-distributing codes is proved. It is also shown that a (k−2)-dense set must contain at least n2/(k−1) points. Equality can be attained if and only if k−1 divides n and there are k−2 orthogonal latin squares of order n/(k−1).
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