{"title":"重新审视线性和圆形单次变更覆盖设计","authors":"Amanda Chafee, Brett Stevens","doi":"10.1002/jcd.21885","DOIUrl":null,"url":null,"abstract":"<p>A <i>single-change covering design</i> (SCCD) is a <math>\n <semantics>\n <mrow>\n <mi>v</mi>\n </mrow>\n <annotation> $v$</annotation>\n </semantics></math>-set <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n </mrow>\n <annotation> $X$</annotation>\n </semantics></math> and an ordered list <math>\n <semantics>\n <mrow>\n <mi>ℒ</mi>\n </mrow>\n <annotation> ${\\rm{ {\\mathcal L} }}$</annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n <mi>b</mi>\n </mrow>\n <annotation> $b$</annotation>\n </semantics></math> blocks of size <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math> where every pair from <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n </mrow>\n <annotation> $X$</annotation>\n </semantics></math> must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given <math>\n <semantics>\n <mrow>\n <mi>v</mi>\n <mo>,</mo>\n <mi>k</mi>\n </mrow>\n <annotation> $v,k$</annotation>\n </semantics></math>. In this paper, we use a new recursive construction to solve the existence of circular SCCD(<math>\n <semantics>\n <mrow>\n <mi>v</mi>\n <mo>,</mo>\n <mn>4</mn>\n <mo>,</mo>\n <mi>b</mi>\n </mrow>\n <annotation> $v,4,b$</annotation>\n </semantics></math>) for all <math>\n <semantics>\n <mrow>\n <mi>v</mi>\n </mrow>\n <annotation> $v$</annotation>\n </semantics></math> and three residue classes of circular SCCD(<math>\n <semantics>\n <mrow>\n <mi>v</mi>\n <mo>,</mo>\n <mn>5</mn>\n <mo>,</mo>\n <mi>b</mi>\n </mrow>\n <annotation> $v,5,b$</annotation>\n </semantics></math>) modulo 16. We solve the existence of three residue classes of SCCD<math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n <mo>,</mo>\n <mn>5</mn>\n <mo>,</mo>\n <mi>b</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $(v,5,b)$</annotation>\n </semantics></math> modulo 16. We prove the existence of circular SCCD<math>\n <semantics>\n <mrow>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>2</mn>\n <mi>c</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <mo>)</mo>\n </mrow>\n <mo>+</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>k</mi>\n <mo>,</mo>\n <msup>\n <mi>c</mi>\n <mn>2</mn>\n </msup>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mn>2</mn>\n <mi>k</mi>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n <mo>)</mo>\n </mrow>\n <mo>+</mo>\n <mi>c</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $(2c(k-1)+1,k,{c}^{2}(2k-2)+c)$</annotation>\n </semantics></math>, for all <math>\n <semantics>\n <mrow>\n <mi>c</mi>\n <mo>≥</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mi>k</mi>\n <mo>≥</mo>\n <mn>2</mn>\n </mrow>\n <annotation> $c\\ge 1,k\\ge 2$</annotation>\n </semantics></math>, using difference methods.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"405-421"},"PeriodicalIF":0.5000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21885","citationCount":"0","resultStr":"{\"title\":\"Linear and circular single-change covering designs revisited\",\"authors\":\"Amanda Chafee, Brett Stevens\",\"doi\":\"10.1002/jcd.21885\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A <i>single-change covering design</i> (SCCD) is a <math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n </mrow>\\n <annotation> $v$</annotation>\\n </semantics></math>-set <math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n </mrow>\\n <annotation> $X$</annotation>\\n </semantics></math> and an ordered list <math>\\n <semantics>\\n <mrow>\\n <mi>ℒ</mi>\\n </mrow>\\n <annotation> ${\\\\rm{ {\\\\mathcal L} }}$</annotation>\\n </semantics></math> of <math>\\n <semantics>\\n <mrow>\\n <mi>b</mi>\\n </mrow>\\n <annotation> $b$</annotation>\\n </semantics></math> blocks of size <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math> where every pair from <math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n </mrow>\\n <annotation> $X$</annotation>\\n </semantics></math> must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given <math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n <mo>,</mo>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $v,k$</annotation>\\n </semantics></math>. In this paper, we use a new recursive construction to solve the existence of circular SCCD(<math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n <mo>,</mo>\\n <mn>4</mn>\\n <mo>,</mo>\\n <mi>b</mi>\\n </mrow>\\n <annotation> $v,4,b$</annotation>\\n </semantics></math>) for all <math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n </mrow>\\n <annotation> $v$</annotation>\\n </semantics></math> and three residue classes of circular SCCD(<math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n <mo>,</mo>\\n <mn>5</mn>\\n <mo>,</mo>\\n <mi>b</mi>\\n </mrow>\\n <annotation> $v,5,b$</annotation>\\n </semantics></math>) modulo 16. We solve the existence of three residue classes of SCCD<math>\\n <semantics>\\n <mrow>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n <mo>,</mo>\\n <mn>5</mn>\\n <mo>,</mo>\\n <mi>b</mi>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $(v,5,b)$</annotation>\\n </semantics></math> modulo 16. We prove the existence of circular SCCD<math>\\n <semantics>\\n <mrow>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mn>2</mn>\\n <mi>c</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>k</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <mo>+</mo>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mi>k</mi>\\n <mo>,</mo>\\n <msup>\\n <mi>c</mi>\\n <mn>2</mn>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mn>2</mn>\\n <mi>k</mi>\\n <mo>−</mo>\\n <mn>2</mn>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <mo>+</mo>\\n <mi>c</mi>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $(2c(k-1)+1,k,{c}^{2}(2k-2)+c)$</annotation>\\n </semantics></math>, for all <math>\\n <semantics>\\n <mrow>\\n <mi>c</mi>\\n <mo>≥</mo>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mi>k</mi>\\n <mo>≥</mo>\\n <mn>2</mn>\\n </mrow>\\n <annotation> $c\\\\ge 1,k\\\\ge 2$</annotation>\\n </semantics></math>, using difference methods.</p>\",\"PeriodicalId\":15389,\"journal\":{\"name\":\"Journal of Combinatorial Designs\",\"volume\":\"31 9\",\"pages\":\"405-421\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21885\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Designs\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21885\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21885","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Linear and circular single-change covering designs revisited
A single-change covering design (SCCD) is a -set and an ordered list of blocks of size where every pair from must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given . In this paper, we use a new recursive construction to solve the existence of circular SCCD() for all and three residue classes of circular SCCD() modulo 16. We solve the existence of three residue classes of SCCD modulo 16. We prove the existence of circular SCCD, for all , using difference methods.
期刊介绍：
The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including:
block designs, t-designs, pairwise balanced designs and group divisible designs
Latin squares, quasigroups, and related algebras
computational methods in design theory
construction methods
applications in computer science, experimental design theory, and coding theory
graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics
finite geometry and its relation with design theory.
algebraic aspects of design theory.
Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.
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