Journal of Combinatorial Designs最新文献
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Enumeration and Construction of Row-Column Designs
行-列设计的枚举和构造
IF 0.5
4区 数学
Gerold Jäger, Klas Markström, Denys Shcherbak, Lars-Daniel Öhman
{"title":"Enumeration and Construction of Row-Column Designs","authors":"Gerold Jäger, Klas Markström, Denys Shcherbak, Lars-Daniel Öhman","doi":"10.1002/jcd.21991","DOIUrl":"https://doi.org/10.1002/jcd.21991","url":null,"abstract":"<p>We computationally completely enumerate a number of types of row-column designs up to isotopism, including double, sesqui, and triple arrays as known from the literature, and two newly introduced types that we call mono arrays and AO-arrays. We calculate autotopism group sizes for the designs we generate. For larger parameter values, where complete enumeration is not feasible, we generate examples of some of the designs, and for some admissible parameter sets, we prove non-existence results. We give some explicit constructions of sesqui arrays, mono arrays and AO-arrays, in particular, we prove constructively that AO-arrays exist for all feasible parameter sets. Finally, we investigate connections to Youden rectangles and binary pseudo-Youden designs.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 9","pages":"357-372"},"PeriodicalIF":0.5,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21991","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Strong External Difference Families and Classification of α -Valuations
强外部差分族与α -估值分类
IF 0.5
4区 数学
Donald L. Kreher, Maura B. Paterson, Douglas R. Stinson
{"title":"Strong External Difference Families and Classification of \u0000 \u0000 α\u0000 -Valuations","authors":"Donald L. Kreher, Maura B. Paterson, Douglas R. Stinson","doi":"10.1002/jcd.21985","DOIUrl":"https://doi.org/10.1002/jcd.21985","url":null,"abstract":"<p>One method of constructing <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>a</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math>-SEDFs (i.e., strong external difference families) in <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>a</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow></math> makes use of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>α</mi>\u0000 </mrow></math>-valuations of complete bipartite graphs <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>a</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow></math>. We explore this approach and we provide a classification theorem which shows that all such <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>α</mi>\u0000 </mrow></math>-valuations can be constructed recursively via a sequence of “blow-up” operations. We also enumerate all <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>a</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 9","pages":"343-356"},"PeriodicalIF":0.5,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21985","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Toward a Baranyai Theorem With an Additional Condition
关于一个附加条件的Baranyai定理
IF 0.5
4区 数学
Gyula O. H. Katona, Gyula Y. Katona
{"title":"Toward a Baranyai Theorem With an Additional Condition","authors":"Gyula O. H. Katona, Gyula Y. Katona","doi":"10.1002/jcd.21992","DOIUrl":"https://doi.org/10.1002/jcd.21992","url":null,"abstract":"<div>\u0000 \u0000 <p>A <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> partial partition of an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-element set is a collection of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> pairwise disjoint <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-element subsets. It is proved that, if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is large enough, one can find <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfenced>\u0000 <mrow>\u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>n</mi>\u0000 \u0000 <mi>k</mi>\u0000 </mfrac>\u0000 </mfenced>\u0000 \u0000 <mo>∕</mo>\u0000 \u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> such partial partitions in such a way that if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 9","pages":"373-376"},"PeriodicalIF":0.5,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Note on Finding Large Transversals Efficiently
关于高效查找大截线的注释
IF 0.5
4区 数学
Michael Anastos, Patrick Morris
{"title":"A Note on Finding Large Transversals Efficiently","authors":"Michael Anastos, Patrick Morris","doi":"10.1002/jcd.21990","DOIUrl":"https://doi.org/10.1002/jcd.21990","url":null,"abstract":"<div>\u0000 \u0000 <p>In an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> array filled with symbols, a <i>transversal</i> is a collection of entries with distinct rows, columns and symbols. In this note we show that if no symbol appears more than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>β</mi>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> times, the array contains a transversal of size <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>1</mn>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>β</mi>\u0000 \u0000 <mo>∕</mo>\u0000 \u0000 <mn>4</mn>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>o</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. In particular, if the array is filled with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> symbols, each appearing <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> times (an equi-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> square), we get transversals of size <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 9","pages":"338-342"},"PeriodicalIF":0.5,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Toward a Solution of Archdeacon's Conjecture on Integer Heffter Arrays
关于整数Heffter数组上Archdeacon猜想的一个解
IF 0.5
4区 数学
Marco Antonio Pellegrini, Tommaso Traetta
{"title":"Toward a Solution of Archdeacon's Conjecture on Integer Heffter Arrays","authors":"Marco Antonio Pellegrini, Tommaso Traetta","doi":"10.1002/jcd.21983","DOIUrl":"https://doi.org/10.1002/jcd.21983","url":null,"abstract":"<p>In this article, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>;</mo>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> whenever the necessary conditions hold, that is, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>0</mn>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 8","pages":"310-323"},"PeriodicalIF":0.5,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21983","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of Magic Rectangle Sets Over Finite Abelian Groups
有限阿贝尔群上幻矩形集的存在性
IF 0.5
4区 数学
Shikang Yu, Tao Feng, Hengrui Liu
{"title":"Existence of Magic Rectangle Sets Over Finite Abelian Groups","authors":"Shikang Yu, Tao Feng, Hengrui Liu","doi":"10.1002/jcd.21987","DOIUrl":"https://doi.org/10.1002/jcd.21987","url":null,"abstract":"<div>\u0000 \u0000 <p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>b</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be positive integers. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mo>+</mo>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a finite abelian group of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mi>b</mi>\u0000 \u0000 <mi>c</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. A <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-magic rectangle set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mstyle>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>MRS</mtext>\u0000 <mspace></mspace>\u0000 </mstyle>\u0000 \u0000 <mi>G</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>b</mi>\u0000 \u0000 <mo>;</mo>\u0000 \u0000 <mi>c</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is a collection of <span></span><math>\u0000 <semantics>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 9","pages":"329-337"},"PeriodicalIF":0.5,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
How to Burn a Latin Square
如何燃烧拉丁广场
IF 0.5
4区 数学
Anthony Bonato, Caleb Jones, Trent G. Marbach, Teddy Mishura
{"title":"How to Burn a Latin Square","authors":"Anthony Bonato, Caleb Jones, Trent G. Marbach, Teddy Mishura","doi":"10.1002/jcd.21988","DOIUrl":"https://doi.org/10.1002/jcd.21988","url":null,"abstract":"<p>We investigate the lazy burning process for Latin squares by studying their associated hypergraphs. In lazy burning, a set of vertices in a hypergraph is initially burned, and that burning spreads to neighboring vertices over time via a specified propagation rule. The lazy burning number is the minimum number of initially burned vertices that eventually burns all vertices. The hypergraphs associated with Latin squares include the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-uniform hypergraph, whose vertices and hyperedges correspond to the entries and lines (i.e., sets of rows, columns, or symbols) of the Latin square, respectively, and the 3-uniform hypergraph, which has vertices corresponding to the lines of the Latin square and hyperedges induced by its entries. Using sequences of vertices that together form a vertex cover, we show that for a Latin square of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, the lazy burning number of its <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-uniform hypergraph is bounded below by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>n</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and above by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>n</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>+</mo>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 8","pages":"300-309"},"PeriodicalIF":0.5,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21988","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Constructions of Optimal Sparse r -Disjunct Matrices via Packings
通过填充构造最优稀疏r -不相交矩阵
IF 0.5
4区 数学
Liying Yu, Xin Wang, Lijun Ji
{"title":"Constructions of Optimal Sparse \u0000 \u0000 \u0000 \u0000 r\u0000 \u0000 \u0000 -Disjunct Matrices via Packings","authors":"Liying Yu, Xin Wang, Lijun Ji","doi":"10.1002/jcd.21986","DOIUrl":"https://doi.org/10.1002/jcd.21986","url":null,"abstract":"<div>\u0000 \u0000 <p>Group testing has been widely used in various aspects, and the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-disjunct matrix plays a crucial role in group testing. The original purpose of the group testing is to identify a set of at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> positive items from a batch of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> total items using as fewer tests as possible. In many practical applications, each test can include only a limited number of items and each item can participate in a limited number of tests. In this paper, we use the tools from combinatorial design theory to construct optimal 2-disjunct matrices with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> rows and limited row weight <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo><</mo>\u0000 \u0000 <mi>ρ</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mrow>\u0000 <mo>⌊</mo>\u0000 \u0000 <mfrac>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 \u0000 <mo>⌋</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and optimal 3-disjunct matrices with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> rows and limited row weight <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>4</mn>\u0000 \u0000 <mo><</mo>\u0000 \u0000 <mi>ρ</mi>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 8","pages":"287-299"},"PeriodicalIF":0.5,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Doubly Orthogonal Equi-Squares and Sliced Orthogonal Arrays
IF 0.5
4区 数学
John Lorch
{"title":"Doubly Orthogonal Equi-Squares and Sliced Orthogonal Arrays","authors":"John Lorch","doi":"10.1002/jcd.21982","DOIUrl":"https://doi.org/10.1002/jcd.21982","url":null,"abstract":"<div>\u0000 \u0000 <p>We introduce doubly orthogonal equi-squares. Using linear algebra over finite fields, we produce large families of mutually <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-doubly orthogonal equi-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>s</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> squares, and show these are of maximal size when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>r</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. These results specialize to the results of Xu, Haaland, and Qian when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and the equi-squares are Sudoku Latin squares of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Further, we show how a collection of mutually <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-doubly orthogonal equi-<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"275-283"},"PeriodicalIF":0.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Terwilliger Algebra of the Group Association Scheme of the Symmetric Group Sym ( 7 )
IF 0.5
4区 数学
Allen Herman, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra
{"title":"On the Terwilliger Algebra of the Group Association Scheme of the Symmetric Group \u0000 \u0000 \u0000 \u0000 Sym\u0000 \u0000 (\u0000 7\u0000 )","authors":"Allen Herman, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1002/jcd.21981","DOIUrl":"https://doi.org/10.1002/jcd.21981","url":null,"abstract":"<p>Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>6</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, have been studied and completely determined. The case for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>7</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is computationally much more difficult and has a potential application to find the size of the largest permutation codes of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>7</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with a minimal distance of at least 4. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"261-274"},"PeriodicalIF":0.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21981","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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