Journal of Combinatorial Designs最新文献_第2页

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

Journal of Combinatorial Designs Pub Date : 2025-06-16 DOI: 10.1002/jcd.21985

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":"One method of constructing <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 <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 <math>\u0000 \u0000 <mrow>\u0000 <mi>α</mi>\u0000 </mrow></math>-valuations of complete bipartite graphs <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 <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 <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

Journal of Combinatorial Designs Pub Date : 2025-06-16 DOI: 10.1002/jcd.21992

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 A <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 <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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> pairwise disjoint <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 <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 <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 <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 <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

Journal of Combinatorial Designs Pub Date : 2025-05-26 DOI: 10.1002/jcd.21990

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 In an <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 transversal is a collection of entries with distinct rows, columns and symbols. In this note we show that if no symbol appears more than <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 <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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> symbols, each appearing <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> times (an equi-<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 <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

Journal of Combinatorial Designs Pub Date : 2025-05-20 DOI: 10.1002/jcd.21983

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":"In this article, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays <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, <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>, <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>, <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 <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

Journal of Combinatorial Designs Pub Date : 2025-05-20 DOI: 10.1002/jcd.21987

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 Let <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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be positive integers. Let <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 <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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-magic rectangle set <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 <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

Journal of Combinatorial Designs Pub Date : 2025-05-15 DOI: 10.1002/jcd.21988

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":"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 <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 <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 <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 <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 <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

Journal of Combinatorial Designs Pub Date : 2025-04-29 DOI: 10.1002/jcd.21986

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 Group testing has been widely used in various aspects, and the <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 <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 <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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> rows and limited row weight <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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> rows and limited row weight <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

Journal of Combinatorial Designs Pub Date : 2025-04-03 DOI: 10.1002/jcd.21982

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 We introduce doubly orthogonal equi-squares. Using linear algebra over finite fields, we produce large families of mutually <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-<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 <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 <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 <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 <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-<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 ) 关于对称群Sym群关联方案的Terwilliger代数(7)

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-04-03 DOI: 10.1002/jcd.21981

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":"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 <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 <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 <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 <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 <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}

引用次数: 0

The Directed Oberwolfach Problem With Variable Cycle Lengths: A Recursive Construction 变周期长的有向Oberwolfach问题：一个递归构造

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-03-24 DOI: 10.1002/jcd.21967

Suzan Kadri, Mateja Šajna

{"title":"The Directed Oberwolfach Problem With Variable Cycle Lengths: A Recursive Construction","authors":"Suzan Kadri, Mateja Šajna","doi":"10.1002/jcd.21967","DOIUrl":"https://doi.org/10.1002/jcd.21967","url":null,"abstract":"The directed Oberwolfach problem <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mstyle>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>OP</mtext>\u0000 <mspace></mspace>\u0000 </mstyle>\u0000 \u0000 <mo>*</mo>\u0000 </msup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>m</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>m</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> asks whether the complete symmetric digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msubsup>\u0000 <mi>K</mi>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>*</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, assuming <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <msub>\u0000 <mi>m</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>⋯</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <msub>\u0000 <mi>m</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, admits a decomposition into spanning subdigraphs, each a disjoint un","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"239-260"},"PeriodicalIF":0.5,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21967","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143945026","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