Journal of Combinatorial Designs最新文献

{"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}

{"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}

{"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}

{"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}

{"title":"Characterising ovoidal cones by their hyperplane intersection numbers","authors":"Bart De Bruyn,&nbsp;Geertrui Van de Voorde","doi":"10.1002/jcd.21959","DOIUrl":"https://doi.org/10.1002/jcd.21959","url":null,"abstract":"<p>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 <span></span><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 <span></span><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>, <span></span><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 <span></span><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.</p>","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}

{"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":"<p>An internal or friendly partition of a vertex set <span></span><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 <span></span><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 <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 <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.</p>","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}

{"title":"Nestings of BIBDs with block size four","authors":"Marco Buratti,&nbsp;Donald L. Kreher,&nbsp;Douglas R. Stinson","doi":"10.1002/jcd.21957","DOIUrl":"https://doi.org/10.1002/jcd.21957","url":null,"abstract":"&lt;p&gt;In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the &lt;i&gt;nested point&lt;/i&gt;) to every block of a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $(v,k,lambda )$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-BIBD in such a way that we end up with a partial &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $(v,k+1,lambda +1)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-BIBD. In the case where the partial &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;λ&lt;/mi&gt;\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}

{"title":"Every latin hypercube of order 5 has transversals","authors":"Alexey L. Perezhogin,&nbsp;Vladimir N. Potapov,&nbsp;Sergey Yu. Vladimirov","doi":"10.1002/jcd.21954","DOIUrl":"10.1002/jcd.21954","url":null,"abstract":"&lt;p&gt;We prove that for all &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $ngt 1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; every latin &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-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 &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $nge 3$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $qge 3$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; we construct a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;⋯&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $(2q-2)times qtimes {rm{cdots }}times q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; latin &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional cuboid of order &lt;span&gt;&lt;/span&gt;&lt;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}

{"title":"Generalised evasive subspaces","authors":"Anina Gruica,&nbsp;Alberto Ravagnani,&nbsp;John Sheekey,&nbsp;Ferdinando Zullo","doi":"10.1002/jcd.21953","DOIUrl":"10.1002/jcd.21953","url":null,"abstract":"<p>We introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a nonconstructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank-metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank-metric codes.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 11","pages":"642-678"},"PeriodicalIF":0.5,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574803","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}

{"title":"On eigenfunctions of the block graphs of geometric Steiner systems","authors":"Sergey Goryainov,&nbsp;Dmitry Panasenko","doi":"10.1002/jcd.21951","DOIUrl":"10.1002/jcd.21951","url":null,"abstract":"<p>This paper lies in the context of the studies of eigenfunctions of graphs having minimum cardinality of support. One of the tools is the weight-distribution bound, a lower bound on the cardinality of support of an eigenfunction of a distance-regular graph corresponding to a nonprincipal eigenvalue. The tightness of the weight-distribution bound was previously shown in general for the smallest eigenvalue of a Grassmann graph. However, a characterisation of optimal eigenfunctions was not obtained. Motivated by this open problem, we consider the class of strongly regular Grassmann graphs and give the required characterisation in this case. We then show the tightness of the weight-distribution bound for block graphs of affine designs (defined on the lines of an affine space with two lines being adjacent when intersect) and obtain a similar characterisation of optimal eigenfunctions.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 11","pages":"629-641"},"PeriodicalIF":0.5,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502912","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}