Journal of Combinatorial Designs最新文献

{"title":"Self-dual association schemes, fusions of Hamming schemes, and partial geometric designs","authors":"Bangteng Xu","doi":"10.1002/jcd.21889","DOIUrl":"https://doi.org/10.1002/jcd.21889","url":null,"abstract":"<p>Partial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self-dual association schemes. Then using the characterizations of self-dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self-dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2-designs. In particular, for a primitive self-dual symmetric association scheme <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>i</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${mathscr{X}}=(X,{{{R}_{i}}}_{0le ile 3})$</annotation>\u0000 </semantics></math> of rank 4, if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>X</mi>\u0000 \u0000 <mo>∣</mo>\u0000 </mrow>\u0000 <annotation> $| X| $</annotation>\u0000 </semantics></math> is a power of 3 and each of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"373-399"},"PeriodicalIF":0.7,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50120463","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":"Primitive \u0000 \u0000 \u0000 \u0000 C\u0000 4\u0000 \u0000 \u0000 ${C}_{4}$\u0000 -decompositions of \u0000 \u0000 \u0000 \u0000 K\u0000 n\u0000 \u0000 −\u0000 I\u0000 \u0000 ${K}_{n}-I$","authors":"Michael W. Schroeder","doi":"10.1002/jcd.21887","DOIUrl":"https://doi.org/10.1002/jcd.21887","url":null,"abstract":"<p>A decomposition <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{C}}$</annotation>\u0000 </semantics></math> of a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is primitive if no proper, nontrivial subset of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{C}}$</annotation>\u0000 </semantics></math> is a decomposition of an induced subgraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. An unresolved question posed by Asplund et al. in a recent publication involves the existence of primitive decompositions of cocktail party graphs into cycles of length 4, which is resolved by this paper.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"368-372"},"PeriodicalIF":0.7,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50122041","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":"Magic partially filled arrays on abelian groups","authors":"Fiorenza Morini,&nbsp;Marco Antonio Pellegrini","doi":"10.1002/jcd.21886","DOIUrl":"https://doi.org/10.1002/jcd.21886","url":null,"abstract":"&lt;p&gt;In this paper we introduce a special class of partially filled arrays. A magic partially filled array &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mtext&gt;MPF&lt;/mtext&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;;&lt;/mo&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${text{MPF}}_{{rm{Omega }}}(m,n;s,k)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on a subset &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{Omega }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of an abelian group &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $({rm{Gamma }},+)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a partially filled array of size &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $mtimes n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with entries in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{Omega }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that (i) every &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $omega in {rm{Omega }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; appears once in the array; (ii) each row contains &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; filled cells and each column contains &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; filled cells; (iii) there exist (not necessarily distinct) elements &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;y&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"347-367"},"PeriodicalIF":0.7,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21886","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50120677","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":"On an Assmus–Mattson type theorem for type I and even formally self-dual codes","authors":"Tsuyoshi Miezaki,&nbsp;Hiroyuki Nakasora","doi":"10.1002/jcd.21883","DOIUrl":"https://doi.org/10.1002/jcd.21883","url":null,"abstract":"<p>In the present paper, we give an Assmus–Mattson type theorem for near-extremal Type I and even formally self-dual codes. We show the existence of 1-designs or 2-designs for these codes. As a corollary, we prove the uniqueness of a self-orthogonal 2-<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>16</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>6</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>8</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(16,6,8)$</annotation>\u0000 </semantics></math> design.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 7","pages":"335-344"},"PeriodicalIF":0.7,"publicationDate":"2023-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50151560","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":"Ordered covering arrays and upper bounds on covering codes","authors":"André Guerino Castoldi,&nbsp;Emerson L. Monte Carmelo,&nbsp;Lucia Moura,&nbsp;Daniel Panario,&nbsp;Brett Stevens","doi":"10.1002/jcd.21882","DOIUrl":"https://doi.org/10.1002/jcd.21882","url":null,"abstract":"<p>This work shows several direct and recursive constructions of ordered covering arrays (OCAs) using projection, fusion, column augmentation, derivation, concatenation, and Cartesian product. Upper bounds on covering codes in Niederreiter–Rosenbloom–Tsfasman (shorten by NRT) spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets. We give tables comparing the new upper bounds for covering codes to existing ones.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 6","pages":"304-329"},"PeriodicalIF":0.7,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50148437","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":"The existence of \u0000 \u0000 \u0000 λ\u0000 \u0000 $lambda $\u0000 -decomposable super-simple \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 4\u0000 ,\u0000 2\u0000 λ\u0000 \u0000 )\u0000 \u0000 \u0000 $(4,2lambda )$\u0000 -GDDs of type \u0000 \u0000 \u0000 \u0000 g\u0000 u\u0000 \u0000 \u0000 ${g}^{u}$\u0000 with \u0000 \u0000 \u0000 λ\u0000 =\u0000 2\u0000 ,\u0000 4\u0000 \u0000 $lambda =2,4$","authors":"Huangsheng Yu,&nbsp;Jingyuan Chen,&nbsp;R. Julian R. Abel,&nbsp;Dianhua Wu","doi":"10.1002/jcd.21881","DOIUrl":"https://doi.org/10.1002/jcd.21881","url":null,"abstract":"&lt;p&gt;A design is said to be &lt;i&gt;super-simple&lt;/i&gt; if the intersection of any two of its blocks has at most two elements. A design with index &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $tlambda $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is said to be &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $lambda $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-&lt;i&gt;decomposable&lt;/i&gt;, if its blocks can be partitioned into nonempty collections &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℬ&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${{rm{ {mathcal B} }}}_{i}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;≤&lt;/mo&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mo&gt;≤&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $1le ile t$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, such that each &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℬ&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${{rm{ {mathcal B} }}}_{i}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with the point set forms a design with index &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $lambda $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this paper, it is proved that for &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $lambda in {2,4}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, there exists a &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $lambda $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-decomposable super-simple &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $(4,2lambda )$&lt;/annotation&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 6","pages":"289-303"},"PeriodicalIF":0.7,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50148436","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":"Towards the Ryser–Woodall \u0000 \u0000 \u0000 λ\u0000 \u0000 $lambda $\u0000 -design conjecture","authors":"Navin M. Singhi,&nbsp;Mohan S. Shrikhande,&nbsp;Rajendra M. Pawale","doi":"10.1002/jcd.21878","DOIUrl":"https://doi.org/10.1002/jcd.21878","url":null,"abstract":"&lt;p&gt;Let &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${r}_{1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${r}_{2},({r}_{1}gt {r}_{2})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the two replication numbers of a &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $lambda $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-design &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $D$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We denote the block size &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;∣&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;∣&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $| {B}_{j}| $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; by &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${k}_{j}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and by &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${k}_{j}^{^{prime} }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (respectively, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 5","pages":"267-276"},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50144158","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":"Alternating groups and point-primitive linear spaces with number of points being squarefree","authors":"Haiyan Guan,&nbsp;Shenglin Zhou","doi":"10.1002/jcd.21879","DOIUrl":"https://doi.org/10.1002/jcd.21879","url":null,"abstract":"<p>This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math> be a nontrivial finite regular linear space whose number of points <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math> is squarefree. We prove that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>≤</mo>\u0000 <mtext>Aut</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $Gle text{Aut}({mathscr{S}})$</annotation>\u0000 </semantics></math> is point-primitive with an alternating socle, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math> is the projective space <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(3,2)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 5","pages":"277-286"},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50144159","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 the chromatic number of some generalized Kneser graphs","authors":"Jozefien D'haeseleer,&nbsp;Klaus Metsch,&nbsp;Daniel Werner","doi":"10.1002/jcd.21875","DOIUrl":"https://doi.org/10.1002/jcd.21875","url":null,"abstract":"&lt;p&gt;We determine the chromatic number of the Kneser graph &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;7&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;4&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;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $q{{rm{Gamma }}}_{7,{3,4}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of flags of vectorial type &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;4&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;annotation&gt; ${3,4}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of a rank 7 vector space over the finite field &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;GF&lt;/mi&gt;\u0000 &lt;mrow&gt;\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 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $mathrm{GF}(q)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for large &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all &lt;i&gt;large&lt;/i&gt; independent sets. Furthermore, our proof is more general in the following sense: it provides the chromatic number of the Kneser graphs &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;d&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 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 4","pages":"179-204"},"PeriodicalIF":0.7,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50127549","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":"Balanced covering arrays: A classification of covering arrays and packing arrays via exact methods","authors":"Ludwig Kampel,&nbsp;Irene Hiess,&nbsp;Ilias S. Kotsireas,&nbsp;Dimitris E. Simos","doi":"10.1002/jcd.21876","DOIUrl":"https://doi.org/10.1002/jcd.21876","url":null,"abstract":"<p>In this paper we investigate the intersections of classes of covering arrays (CAs) and packing arrays (PAs). The arrays appearing in these intersections obey to upper and lower bounds regarding the appearance of tuples in sub-matrices—we call these arrays <i>balanced covering arrays</i>. We formulate and formalize first observations for which upper and lower bounds on the appearance of tuples it is of interest to consider these intersections of CAs and PAs. Outside of these bounds the intersections will be either empty, for the case of too restrictive constraints, or equal to the maximum element in the emerging lattices, for the case of too weak constraints. We present a column extension algorithm for classification of nonequivalent balanced CAs that uses a SAT solver or a pseudo-Boolean (PB) solver to compute the columns suitable for array extension together with a lex-leader ordering to identify unique representatives for each equivalence class of balanced CAs. These computations bring to light a dissection of classes of CAs that is partially nested due to the nature of the considered intersections. These dissections can be trivial, containing only a single type of balanced CAs, or can also appear as highly structured containing multiple nested types of balanced CAs. Our results indicate that balanced CAs are an interesting class of designs that is rich of structure.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 4","pages":"205-261"},"PeriodicalIF":0.7,"publicationDate":"2023-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50121488","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}