Journal of Combinatorial Designs最新文献_第2页

On Quasi-Hermitian Varieties in Even Characteristic and Related Orthogonal Arrays

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-01-06 DOI: 10.1002/jcd.21966

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":"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 <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 <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 <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.","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

Journal of Combinatorial Designs Pub Date : 2024-12-30 DOI: 10.1002/jcd.21968

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 Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a prime power and <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 <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 <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 <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 <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 <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

Journal of Combinatorial Designs Pub Date : 2024-11-17 DOI: 10.1002/jcd.21961

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":"We consider playing the game of Tic-Tac-Toe on block designs BIBD<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<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<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 <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 <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

Journal of Combinatorial Designs Pub Date : 2024-11-17 DOI: 10.1002/jcd.21962

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 There are four nonisomorphic configurations of triples that can form a triangle in a three-uniform hypergraph, where the configurations <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>B</mi>\u0000 </mrow>\u0000 <annotation> ${bf{B}}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> ${bf{D}}$</annotation>\u0000 </semantics></math> on <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 <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 <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<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<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 <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

Journal of Combinatorial Designs Pub Date : 2024-10-16 DOI: 10.1002/jcd.21960

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":"Given an integer partition <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 <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 <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 <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 <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 <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

Journal of Combinatorial Designs Pub Date : 2024-10-10 DOI: 10.1002/jcd.21958

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":"A design is said to be super-simple if the intersection of any two blocks has at most two elements. A design with index <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math> is said to be completely reducible, if its blocks can be partitioned into nonempty collections <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 <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) <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 (<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 <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}

引用次数: 0

Characterising ovoidal cones by their hyperplane intersection numbers 通过超平面相交数确定卵圆锥的特征

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-10-09 DOI: 10.1002/jcd.21959

Bart De Bruyn, Geertrui Van de Voorde

{"title":"Characterising ovoidal cones by their hyperplane intersection numbers","authors":"Bart De Bruyn, Geertrui Van de Voorde","doi":"10.1002/jcd.21959","DOIUrl":"https://doi.org/10.1002/jcd.21959","url":null,"abstract":"In this paper, we characterise point sets having the same intersection numbers with respect to hyperplanes as an ovoidal cone. In particular, we show that a set of points of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(4,q)$</annotation>\u0000 </semantics></math> which blocks all planes and intersects solids in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $q+1$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> ${q}^{2}+1$</annotation>\u0000 </semantics></math> or <math>\u0000 <semantics>\u0000 <mrow>\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 <annotation> ${q}^{2}+q+1$</annotation>\u0000 </semantics></math> points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 1","pages":"5-26"},"PeriodicalIF":0.5,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21959","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142665775","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

Partitioning the projective plane into two incidence-rich parts 将投影面划分为两个入射丰富的部分

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-10-06 DOI: 10.1002/jcd.21956

Zoltán Lóránt Nagy

{"title":"Partitioning the projective plane into two incidence-rich parts","authors":"Zoltán Lóránt Nagy","doi":"10.1002/jcd.21956","DOIUrl":"https://doi.org/10.1002/jcd.21956","url":null,"abstract":"An internal or friendly partition of a vertex set <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $V(G)$</annotation>\u0000 </semantics></math> of a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a partition to two nonempty sets <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 <annotation> $Acup B$</annotation>\u0000 </semantics></math> such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbour set in its own class and the neighbour set of the other class that can be attained for all vertices at the same time for the incidence graphs of Desarguesian planes of square order.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 12","pages":"703-714"},"PeriodicalIF":0.5,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21956","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142429303","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

Nestings of BIBDs with block size four 块大小为 4 的 BIBD 嵌套

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-10-06 DOI: 10.1002/jcd.21957

Marco Buratti, Donald L. Kreher, Douglas R. Stinson

{"title":"Nestings of BIBDs with block size four","authors":"Marco Buratti, Donald L. Kreher, Douglas R. Stinson","doi":"10.1002/jcd.21957","DOIUrl":"https://doi.org/10.1002/jcd.21957","url":null,"abstract":"In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a <math>\u0000 <semantics>\u0000 <mrow>\u0000 \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 </mrow>\u0000 <annotation> $(v,k,lambda )$</annotation>\u0000 </semantics></math>-BIBD in such a way that we end up with a partial <math>\u0000 <semantics>\u0000 <mrow>\u0000 \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 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v,k+1,lambda +1)$</annotation>\u0000 </semantics></math>-BIBD. In the case where the partial <math>\u0000 <semantics>\u0000 <mrow>\u0000 \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 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 12","pages":"715-743"},"PeriodicalIF":0.5,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21957","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142429304","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

Every latin hypercube of order 5 has transversals 每个 5 阶拉丁超立方体都有横轴

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2024-07-30 DOI: 10.1002/jcd.21954

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

{"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":"10.1002/jcd.21954","url":null,"abstract":"We prove that for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $ngt 1$</annotation>\u0000 </semantics></math> every latin <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </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>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $nge 3$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $qge 3$</annotation>\u0000 </semantics></math> we construct a <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>⋯</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(2q-2)times qtimes {rm{cdots }}times q$</annotation>\u0000 </semantics></math> latin <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-dimensional cuboid of order <ma","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 11","pages":"679-699"},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863809","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