每个 5 阶拉丁超立方体都有横轴

Pub Date : 2024-07-30 DOI:10.1002/jcd.21954

Alexey L. Perezhogin, Vladimir N. Potapov, Sergey Yu. Vladimirov

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For each <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> $n\\ge 3$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>q</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> $q\\ge 3$</annotation>\n </semantics></math> we construct a <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mn>2</mn>\n \n <mi>q</mi>\n \n <mo>−</mo>\n \n <mn>2</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>×</mo>\n \n <mi>q</mi>\n \n <mo>×</mo>\n \n <mi>⋯</mi>\n \n <mo>×</mo>\n \n <mi>q</mi>\n </mrow>\n </mrow>\n <annotation> $(2q-2)\\times q\\times {\\rm{\\cdots }}\\times q$</annotation>\n </semantics></math> latin <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>-dimensional cuboid of order <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>q</mi>\n </mrow>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math> with no transversals. Moreover, we find all paratopy classes of nonextendible and noncompletable latin cuboids of order 5.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Every latin hypercube of order 5 has transversals\",\"authors\":\"Alexey L. Perezhogin, Vladimir N. Potapov, Sergey Yu. Vladimirov\",\"doi\":\"10.1002/jcd.21954\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove that for all <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>></mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </mrow>\\n <annotation> $n\\\\gt 1$</annotation>\\n </semantics></math> every latin <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math>-dimensional cube of order 5 has transversals. We find all 123 paratopy classes of layer-latin cubes of order 5 with no transversals. For each <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>≥</mo>\\n \\n <mn>3</mn>\\n </mrow>\\n </mrow>\\n <annotation> $n\\\\ge 3$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>q</mi>\\n \\n <mo>≥</mo>\\n \\n <mn>3</mn>\\n </mrow>\\n </mrow>\\n <annotation> $q\\\\ge 3$</annotation>\\n </semantics></math> we construct a <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>q</mi>\\n \\n <mo>−</mo>\\n \\n <mn>2</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>×</mo>\\n \\n <mi>q</mi>\\n \\n <mo>×</mo>\\n \\n <mi>⋯</mi>\\n \\n <mo>×</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n </mrow>\\n <annotation> $(2q-2)\\\\times q\\\\times {\\\\rm{\\\\cdots }}\\\\times q$</annotation>\\n </semantics></math> latin <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math>-dimensional cuboid of order <math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>q</mi>\\n </mrow>\\n </mrow>\\n <annotation> $q$</annotation>\\n </semantics></math> with no transversals. Moreover, we find all paratopy classes of nonextendible and noncompletable latin cuboids of order 5.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21954\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21954","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

我们证明了所有阶为 5 的层拉丁立方体都有横轴。我们找到了所有 123 个无横轴的 5 阶拉丁层立方体的准类。对于每个且，我们都构造了一个无横轴的阶拉丁立方体。此外，我们还找到了所有阶为 5 的不可扩展和不可完成的拉丁立方体的准类。

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Every latin hypercube of order 5 has transversals

We prove that for all $n > 1$ every latin $n$ -dimensional cube of order 5 has transversals. We find all 123 paratopy classes of layer-latin cubes of order 5 with no transversals. For each $n \geq 3$ and $q \geq 3$ we construct a $(2 q - 2) \times q \times \dots \times q$ latin $n$ -dimensional cuboid of order $q$ with no transversals. Moreover, we find all paratopy classes of nonextendible and noncompletable latin cuboids of order 5.

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