Journal of Combinatorial Designs最新文献

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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":"<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}
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,&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}
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":"<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}
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,&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}
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,&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}
引用次数: 0
Generalised evasive subspaces 广义回避子空间
IF 0.5 4区 数学
Journal of Combinatorial Designs Pub Date : 2024-07-08 DOI: 10.1002/jcd.21953
Anina Gruica, Alberto Ravagnani, John Sheekey, Ferdinando Zullo
{"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}
引用次数: 0
On eigenfunctions of the block graphs of geometric Steiner systems 论几何斯坦纳系统块图的特征函数
IF 0.5 4区 数学
Journal of Combinatorial Designs Pub Date : 2024-06-24 DOI: 10.1002/jcd.21951
Sergey Goryainov, Dmitry Panasenko
{"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}
引用次数: 0
Symmetric 2- ( 36 , 15 , 6 ) $(36,15,6)$ designs with an automorphism of order two 对称 2- ( 36 , 15 , 6 ) $(36,15,6)$设计的二阶自变量
IF 0.5 4区 数学
Journal of Combinatorial Designs Pub Date : 2024-06-17 DOI: 10.1002/jcd.21952
Sanja Rukavina, Vladimir D. Tonchev
{"title":"Symmetric 2-\u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 36\u0000 ,\u0000 15\u0000 ,\u0000 6\u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 $(36,15,6)$\u0000 designs with an automorphism of order two","authors":"Sanja Rukavina,&nbsp;Vladimir D. Tonchev","doi":"10.1002/jcd.21952","DOIUrl":"https://doi.org/10.1002/jcd.21952","url":null,"abstract":"&lt;p&gt;Bouyukliev, Fack and Winne classified all 2-&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;36&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;15&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;6&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; $(36,15,6)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2-&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;36&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;15&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;6&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; $(36,15,6)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; designs that admit an automorphism of order two. It is shown that there are exactly &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;547&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;701&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $1,547,701$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; nonisomorphic such designs, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;135&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;779&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $135,779$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of which are self-dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near-extremal self-dual cod","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 10","pages":"606-624"},"PeriodicalIF":0.5,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967981","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
Mutual incidence matrix of two balanced incomplete block designs 两个平衡不完全区块设计的互现矩阵
IF 0.5 4区 数学
Journal of Combinatorial Designs Pub Date : 2024-06-17 DOI: 10.1002/jcd.21949
Alexander Shramchenko, Vasilisa Shramchenko
{"title":"Mutual incidence matrix of two balanced incomplete block designs","authors":"Alexander Shramchenko,&nbsp;Vasilisa Shramchenko","doi":"10.1002/jcd.21949","DOIUrl":"https://doi.org/10.1002/jcd.21949","url":null,"abstract":"<p>We propose to consider a mutual incidence matrix <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $M$</annotation>\u0000 </semantics></math> of two balanced incomplete block designs built on the same finite set. In the simplest case, this matrix reduces to the standard incidence matrix of one block design. We find all eigenvalues of the matrices <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>M</mi>\u0000 \u0000 <msup>\u0000 <mi>M</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $M{M}^{T}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>M</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msup>\u0000 \u0000 <mi>M</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${M}^{T}M$</annotation>\u0000 </semantics></math> and their eigenspaces.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 10","pages":"579-590"},"PeriodicalIF":0.5,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21949","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967979","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
Infinite series of 3-designs in the extended quadratic residue code 扩展二次残差码中的 3-设计无限序列
IF 0.5 4区 数学
Journal of Combinatorial Designs Pub Date : 2024-06-17 DOI: 10.1002/jcd.21950
Madoka Awada
{"title":"Infinite series of 3-designs in the extended quadratic residue code","authors":"Madoka Awada","doi":"10.1002/jcd.21950","DOIUrl":"https://doi.org/10.1002/jcd.21950","url":null,"abstract":"<p>In this paper, we show an infinite series of 3-designs in the extended quadratic residue codes over <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <msup>\u0000 <mi>r</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{{r}^{2}}$</annotation>\u0000 </semantics></math> for a prime <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 10","pages":"591-605"},"PeriodicalIF":0.5,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967980","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
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