{"title":"组合设计马赛克小实例","authors":"Vedran Krčadinac","doi":"10.1016/j.exco.2024.100163","DOIUrl":null,"url":null,"abstract":"<div><div>We give the first example of a mosaic of three combinatorial designs with distinct parameters 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>, 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>, and 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></math></span>. Furthermore, we give examples of mosaics of 2-<span><math><mrow><mo>(</mo><mn>9</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span> designs that are not resolvable, thereby answering a question posed by M. Wiese and H. Boche. Finally, we give an example of a mosaic of projective planes of order 3 that cannot be obtained by tiling groups with difference sets.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100163"},"PeriodicalIF":0.0000,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Small examples of mosaics of combinatorial designs\",\"authors\":\"Vedran Krčadinac\",\"doi\":\"10.1016/j.exco.2024.100163\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We give the first example of a mosaic of three combinatorial designs with distinct parameters 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>, 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>, and 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></math></span>. Furthermore, we give examples of mosaics of 2-<span><math><mrow><mo>(</mo><mn>9</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span> designs that are not resolvable, thereby answering a question posed by M. Wiese and H. Boche. Finally, we give an example of a mosaic of projective planes of order 3 that cannot be obtained by tiling groups with difference sets.</div></div>\",\"PeriodicalId\":100517,\"journal\":{\"name\":\"Examples and Counterexamples\",\"volume\":\"6 \",\"pages\":\"Article 100163\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-10-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Examples and Counterexamples\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2666657X24000296\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Examples and Counterexamples","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2666657X24000296","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们首次给出了三个组合设计的马赛克的例子，这三个组合设计的参数分别为 2-(13,3,1)、2-(13,4,2) 和 2-(13,6,5)。此外，我们还举例说明了不可解的 2-(9,3,2) 设计的马赛克，从而回答了 M. Wiese 和 H. Boche 提出的问题。最后，我们给出了一个阶数为 3 的投影面马赛克的例子，该马赛克无法通过差集平铺组获得。
Small examples of mosaics of combinatorial designs
We give the first example of a mosaic of three combinatorial designs with distinct parameters 2-, 2-, and 2-. Furthermore, we give examples of mosaics of 2- designs that are not resolvable, thereby answering a question posed by M. Wiese and H. Boche. Finally, we give an example of a mosaic of projective planes of order 3 that cannot be obtained by tiling groups with difference sets.
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