Journal of Combinatorial Designs最新文献
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New Families of Strength-3 Covering Arrays Using Linear Feedback Shift Register Sequences
IF 0.5
4区 数学
Kianoosh Shokri, Lucia Moura
{"title":"New Families of Strength-3 Covering Arrays Using Linear Feedback Shift Register Sequences","authors":"Kianoosh Shokri, Lucia Moura","doi":"10.1002/jcd.21963","DOIUrl":"https://doi.org/10.1002/jcd.21963","url":null,"abstract":"<p>In an array over an alphabet of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> symbols, a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-set of column indices <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>c</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>c</mi>\u0000 \u0000 <mi>t</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is <i>covered</i> if each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-tuple of the alphabet occurs at least once as a row of the sub-array indexed by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>c</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>c</mi>\u0000 \u0000 <mi>t</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. A <i>covering array</i>, denoted by CA<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>N</mi>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 4","pages":"156-171"},"PeriodicalIF":0.5,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21963","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446750","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
On MSRD Codes, h-Designs and Disjoint Maximum Scattered Linear Sets
IF 0.5
4区 数学
Paolo Santonastaso, John Sheekey
{"title":"On MSRD Codes, h-Designs and Disjoint Maximum Scattered Linear Sets","authors":"Paolo Santonastaso, John Sheekey","doi":"10.1002/jcd.21972","DOIUrl":"https://doi.org/10.1002/jcd.21972","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we construct new optimal subspace designs and, consequently, new optimal codes in the sum-rank metric. We construct new 1-designs by finding sets of disjoint maximum scattered linear sets, and use these constructions to also find new <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-designs for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 <mo>></mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. As a means of achieving this, we establish a correspondence between the metric properties of sum-rank metric codes and the geometric properties of subspace designs. Specifically, we determine the geometric counterpart of the coding-theoretic notion of generalised weights for the sum-rank metric in terms of subspace designs and determine a geometric characterisation of MSRD codes. This enables us to characterise subspace designs via their intersections with hyperplanes and via duality operations.</p>\u0000 </div>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 4","pages":"137-155"},"PeriodicalIF":0.5,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143447152","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
Bent Functions on Finite Nonabelian Groups and Relative Difference Sets
IF 0.5
4区 数学
Bangteng Xu
{"title":"Bent Functions on Finite Nonabelian Groups and Relative Difference Sets","authors":"Bangteng Xu","doi":"10.1002/jcd.21970","DOIUrl":"https://doi.org/10.1002/jcd.21970","url":null,"abstract":"<div>\u0000 \u0000 <p>It is well known that the perfect nonlinearity of a function between finite groups <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> can be characterized by its graph in terms of relative difference set in the direct product <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> (cf. [4]). Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be the infinite set of complex roots of unity. A <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-valued function <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> on an arbitrary finite group <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is associated with a finite cyclic subgroup <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mi>f</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in the multiplicative group of nonzero complex numbers. For a bent function <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in general, its graph is not a relative difference set in the direct product <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 4","pages":"125-136"},"PeriodicalIF":0.5,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446694","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
Extensions of Steiner Triple Systems
IF 0.5
4区 数学
Giovanni Falcone, Agota Figula, Mario Galici
{"title":"Extensions of Steiner Triple Systems","authors":"Giovanni Falcone, Agota Figula, Mario Galici","doi":"10.1002/jcd.21964","DOIUrl":"https://doi.org/10.1002/jcd.21964","url":null,"abstract":"<p>In this article, we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"94-108"},"PeriodicalIF":0.5,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21964","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112441","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
On Quasi-Hermitian Varieties in Even Characteristic and Related Orthogonal Arrays
IF 0.5
4区 数学
Angela Aguglia, Luca Giuzzi, Alessandro Montinaro, Viola Siconolfi
{"title":"On Quasi-Hermitian Varieties in Even Characteristic and Related Orthogonal Arrays","authors":"Angela Aguglia, Luca Giuzzi, Alessandro Montinaro, Viola Siconolfi","doi":"10.1002/jcd.21966","DOIUrl":"https://doi.org/10.1002/jcd.21966","url":null,"abstract":"<p>In this article, we study the BM quasi-Hermitian varieties, laying in the three-dimensional Desarguesian projective space of even order. After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent, exhibiting a behavior which is strikingly different from what happens in odd characteristic. This completes the classification project started there. Here we prove more; indeed, by using previous results, we explicitly determine the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct a family of simple orthogonal arrays <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>O</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>5</mn>\u0000 </msup>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msup>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, with entries in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed to investigate how variables in testing interact with each other.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"109-122"},"PeriodicalIF":0.5,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21966","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112442","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
Avoiding Secants of Given Size in Finite Projective Planes
IF 0.5
4区 数学
Tamás Héger, Zoltán Lóránt Nagy
{"title":"Avoiding Secants of Given Size in Finite Projective Planes","authors":"Tamás Héger, Zoltán Lóránt Nagy","doi":"10.1002/jcd.21968","DOIUrl":"https://doi.org/10.1002/jcd.21968","url":null,"abstract":"<div>\u0000 \u0000 <p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a prime power and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a natural number. What are the possible cardinalities of point sets <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in a projective plane of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, which do not intersect any line at exactly <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\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 <mo>∣</mo>\u0000 \u0000 <mi>S</mi>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>=<","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"83-93"},"PeriodicalIF":0.5,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143120778","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
Tic-Tac-Toe on Designs
设计上的井字游戏
IF 0.5
4区 数学
Peter Danziger, Melissa A. Huggan, Rehan Malik, Trent G. Marbach
{"title":"Tic-Tac-Toe on Designs","authors":"Peter Danziger, Melissa A. Huggan, Rehan Malik, Trent G. Marbach","doi":"10.1002/jcd.21961","DOIUrl":"https://doi.org/10.1002/jcd.21961","url":null,"abstract":"<p>We consider playing the game of Tic-Tac-Toe on block designs BIBD<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v,k,lambda )$</annotation>\u0000 </semantics></math> and transversal designs TD<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(k,n)$</annotation>\u0000 </semantics></math>. Players take turns choosing points and the first player to complete a block wins the game. We show that triple systems, BIBD<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v,3,lambda )$</annotation>\u0000 </semantics></math>, are a first-player win if and only if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>5</mn>\u0000 </mrow>\u0000 <annotation> $vge 5$</annotation>\u0000 </semantics></math>. Further, we show that for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 2","pages":"58-71"},"PeriodicalIF":0.5,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21961","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142859908","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
An Improvement on Triple Systems Without Two Types of Configurations
无两种构型的三重系统的改进
IF 0.5
4区 数学
Liying Yu, Shuhui Yu, Lijun Ji
{"title":"An Improvement on Triple Systems Without Two Types of Configurations","authors":"Liying Yu, Shuhui Yu, Lijun Ji","doi":"10.1002/jcd.21962","DOIUrl":"https://doi.org/10.1002/jcd.21962","url":null,"abstract":"<div>\u0000 \u0000 <p>There are four nonisomorphic configurations of triples that can form a triangle in a three-uniform hypergraph, where the configurations <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>B</mi>\u0000 </mrow>\u0000 <annotation> ${bf{B}}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> ${bf{D}}$</annotation>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>5</mn>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation> ${1,2,3,4,5}$</annotation>\u0000 </semantics></math> consist of three triples <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>125</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>134</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>234</mn>\u0000 </mrow>\u0000 <annotation> $125,134,234$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>123</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>134</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>235</mn>\u0000 </mrow>\u0000 <annotation> $123,134,235$</annotation>\u0000 </semantics></math>, respectively. Denote by ex<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>D</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(n,{bf{D}})$</annotation>\u0000 </semantics></math> and ex<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>BD</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(n,{bf{BD}})$</annotation>\u0000 </semantics></math> the maximum number of triples in a three-uniform hypergraph on <span></span><m","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 2","pages":"72-78"},"PeriodicalIF":0.5,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861640","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
Latin squares with five disjoint subsquares
有五个不相交的子正方形的拉丁正方形
IF 0.5
4区 数学
Tara Kemp
{"title":"Latin squares with five disjoint subsquares","authors":"Tara Kemp","doi":"10.1002/jcd.21960","DOIUrl":"https://doi.org/10.1002/jcd.21960","url":null,"abstract":"<p>Given an integer partition <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>h</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <msub>\u0000 <mi>h</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <msub>\u0000 <mi>h</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $({h}_{1}{h}_{2}{rm{ldots }}{h}_{k})$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>, is it possible to find an order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> latin square with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> pairwise disjoint subsquares of orders <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>h</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>h</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${h}_{1},{rm{ldots }},{h}_{k}$</annotation>\u0000 </semantics></math>? This question was posed by Fuchs and has been answered for all partitions with <span></span><math>\u0000 <semantics>\u0000 <mrow","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 2","pages":"39-57"},"PeriodicalIF":0.5,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861593","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
Completely reducible super-simple ( v , 4 , 4 ) $(v,4,4)$ -BIBDs and related constant weight codes
完全还原的超简单 ( v , 4 , 4 ) $(v,4,4)$ -BIBD 及相关恒权码
IF 0.5
4区 数学
Jingyuan Chen, Huangsheng Yu, R. Julian R. Abel, Dianhua Wu
{"title":"Completely reducible super-simple \u0000 \u0000 \u0000 (\u0000 \u0000 v\u0000 ,\u0000 4\u0000 ,\u0000 4\u0000 \u0000 )\u0000 \u0000 $(v,4,4)$\u0000 -BIBDs and related constant weight codes","authors":"Jingyuan Chen, Huangsheng Yu, R. Julian R. Abel, Dianhua Wu","doi":"10.1002/jcd.21958","DOIUrl":"https://doi.org/10.1002/jcd.21958","url":null,"abstract":"<p>A design is said to be <i>super-simple</i> if the intersection of any two blocks has at most two elements. A design with index <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math> is said to be <i>completely reducible</i>, if its blocks can be partitioned into nonempty collections <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>i</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> ${{mathscr{B}}}_{i},1le ile lambda $</annotation>\u0000 </semantics></math>, such that each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathscr{B}}}_{i}$</annotation>\u0000 </semantics></math> together with the point set forms a design with index unity. In this paper, it is proved that there exists a completely reducible super-simple (CRSS) <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(v,4,4)$</annotation>\u0000 </semantics></math> balanced incomplete block design (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(v,4,4)$</annotation>\u0000 </semantics></math>-BIBD for short) if and only if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>13</mn>\u0000 </mrow>\u0000 <annotation> $vge 13$</annotation>\u0000 </semantics></math>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 1","pages":"27-36"},"PeriodicalIF":0.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142665937","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}
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