Journal of Combinatorial Designs最新文献

{"title":"Multifold 1-perfect codes","authors":"Denis S. Krotov","doi":"10.1002/jcd.21947","DOIUrl":"10.1002/jcd.21947","url":null,"abstract":"<p>A multifold 1-perfect code (1-perfect code for list decoding) in any graph is a set <span></span><math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>C</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $C$\u0000</annotation>\u0000 </semantics>\u0000 </math> of vertices such that every vertex of the graph is at distance not more than 1 from exactly <span></span><math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>μ</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $mu $\u0000</annotation>\u0000 </semantics>\u0000 </math> elements of <span></span><math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>C</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $C$\u0000</annotation>\u0000 </semantics>\u0000 </math>. In <span></span><math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $q$\u0000</annotation>\u0000 </semantics>\u0000 </math>-ary Hamming graphs, where <span></span><math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <annotation>\u0000 $q$\u0000</annotation>\u0000 </semantics>\u0000 </math> is a prime power, we characterize all parameters of multifold 1-perfect codes and all parameters of additive multifold 1-perfect codes. In particular, we show that additive multifold 1-perfect codes are related to special multiset generalizations of spreads, multispreads, and that multispreads of parameters corresponding to multifold 1-perfect codes always exist.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 9","pages":"546-555"},"PeriodicalIF":0.5,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147910","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":"Block-transitive triple systems with sporadic or alternating socle","authors":"Suyun Ding,&nbsp;Yilin Zhang,&nbsp;Xiaoqin Zhan,&nbsp;Guangzu Chen","doi":"10.1002/jcd.21945","DOIUrl":"10.1002/jcd.21945","url":null,"abstract":"<p>This paper is a contribution to the classification of all pairs <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $({mathscr{T}},G)$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{T}}$</annotation>\u0000 </semantics></math> is a triple system and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a block-transitive but not flag-transitive automorphism group of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{T}}$</annotation>\u0000 </semantics></math>. We prove that if the socle of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a sporadic or alternating group, then one of the following holds:\u0000\u0000 </p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 9","pages":"521-531"},"PeriodicalIF":0.5,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141121084","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 products of strong Skolem starters","authors":"Oleg Ogandzhanyants,&nbsp;Margarita Kondratieva,&nbsp;Nabil Shalaby","doi":"10.1002/jcd.21943","DOIUrl":"10.1002/jcd.21943","url":null,"abstract":"<p>In 1991, Shalaby conjectured that any <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow></math>, where <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math> or <span></span><math>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mn>8</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>11</mn>\u0000 </mrow></math>, admits a strong Skolem starter. In 2018, the authors fully described and explicitly constructed the infinite “cardioidal” family of strong Skolem starters. No other infinite family of these combinatorial designs was known to date. Statements regarding the products of starters, proven in this paper give a new way of generating strong or skew Skolem starters of composite orders. This approach extends our previous result by generating new infinite families of these starters that are not cardioidal.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"464-487"},"PeriodicalIF":0.7,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942168","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":"New results on large sets of orthogonal arrays and orthogonal arrays","authors":"Guangzhou Chen,&nbsp;Xiaodong Niu,&nbsp;Jiufeng Shi","doi":"10.1002/jcd.21944","DOIUrl":"10.1002/jcd.21944","url":null,"abstract":"<p>Orthogonal array and a large set of orthogonal arrays are important research objects in combinatorial design theory, and they are widely applied to statistics, computer science, coding theory, and cryptography. In this paper, some new series of large sets of orthogonal arrays are given by direct construction, juxtaposition construction, Hadamard construction, finite field construction, and difference matrix construction. Subsequently, many new infinite classes of orthogonal arrays are obtained by using these large sets of orthogonal arrays and Kronecker product.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"488-515"},"PeriodicalIF":0.7,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930827","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 Terwilliger algebras of the group association schemes of three metacyclic groups","authors":"Jing Yang,&nbsp;Xiaoqian Zhang,&nbsp;Lihua Feng","doi":"10.1002/jcd.21941","DOIUrl":"10.1002/jcd.21941","url":null,"abstract":"&lt;p&gt;For any finite group &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt;, the Terwilliger algebra &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; of the group association scheme satisfies the following inclusions: &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&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;T&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;⊆&lt;/mo&gt;\u0000 \u0000 &lt;mover&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;˜&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; is a specific vector space and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;˜&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; is the centralizer algebra of the permutation representation of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; induced by the action of conjugation. The group &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; is said to be triply transitive if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"438-463"},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799847","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 equitably 2-colourable odd cycle decompositions","authors":"Andrea Burgess,&nbsp;Francesca Merola","doi":"10.1002/jcd.21937","DOIUrl":"10.1002/jcd.21937","url":null,"abstract":"&lt;p&gt;An &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt;-cycle decomposition of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; is said to be &lt;i&gt;equitably 2-colourable&lt;/i&gt; if there is a 2-vertex-colouring of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; of the decomposition, each colour appears on &lt;span&gt;&lt;/span&gt;&lt;math&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;ℓ&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 &lt;/mrow&gt;&lt;/math&gt; or &lt;span&gt;&lt;/span&gt;&lt;math&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;ℓ&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 &lt;/mrow&gt;&lt;/math&gt; of the vertices of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt;. In this paper we study the existence of equitably 2-colourable &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt;-cycle decompositions of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt;, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; is odd, and prove the existence of such a decomposition for &lt;span&gt;&lt;/span&gt;&lt;math&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;mn&gt;1&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; (mod &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"419-437"},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21937","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140630988","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":"Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG\u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 3\u0000 ,\u0000 q\u0000 \u0000 )\u0000 \u0000 \u0000 $(3,q)$","authors":"Philipp Heering,&nbsp;Klaus Metsch","doi":"10.1002/jcd.21940","DOIUrl":"10.1002/jcd.21940","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{Gamma }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the graph whose vertices are the chambers of the finite projective 3-space &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;PG&lt;/mtext&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 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;q&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; $text{PG}(3,q)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, with two vertices being adjacent if and only if the corresponding chambers are in general position. We show that a maximal independent set of vertices of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{Gamma }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; contains &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${q}^{4}+3{q}^{3}+4{q}^{2}+3q+1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, or &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $3{q}^{3}+5{q}^{2}+3q+1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, or at most &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 7","pages":"388-409"},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21940","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570787","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 explicit construction for large sets of infinite dimensional \u0000 \u0000 \u0000 q\u0000 \u0000 $q$\u0000 -Steiner systems","authors":"Daniel R. Hawtin","doi":"10.1002/jcd.21942","DOIUrl":"10.1002/jcd.21942","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $V$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a vector space over the finite field &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${{mathbb{F}}}_{q}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. A &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \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;-&lt;i&gt;Steiner system&lt;/i&gt;, or an &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&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;V&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $S{(t,k,V)}_{q}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, is a collection &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℬ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{{mathcal B}}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \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;-dimensional subspaces of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $V$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that every &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $t$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional subspace of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $V$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is contained in a unique element of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 \u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℬ&lt;/mi&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"413-418"},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570622","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":"Genuinely nonabelian partial difference sets","authors":"John Polhill,&nbsp;James A. Davis,&nbsp;Ken W. Smith,&nbsp;Eric Swartz","doi":"10.1002/jcd.21938","DOIUrl":"10.1002/jcd.21938","url":null,"abstract":"<p>Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of <i>genuinely nonabelian</i> PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 7","pages":"351-370"},"PeriodicalIF":0.7,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21938","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570623","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":"Doubly sequenceable groups","authors":"Mohammad Javaheri","doi":"10.1002/jcd.21939","DOIUrl":"10.1002/jcd.21939","url":null,"abstract":"&lt;p&gt;Given a sequence &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&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;msub&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${bf{g}}:{g}_{0},{rm{ldots }},{g}_{m}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in a finite group &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 \u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${g}_{0}={1}_{G}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;¯&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 \u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;¯&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&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;msub&gt;\u0000 &lt;mover&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;¯&lt;/mo&gt;\u0000 &lt;/mover&gt;\u0000 \u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $bar{{bf{g}}}:{bar{g}}_{0},{rm{ldots }},{bar{g}}_{m}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the sequence of consecutive quotients of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${bf{g}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; defined by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mover&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 7","pages":"371-387"},"PeriodicalIF":0.7,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570602","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}