Journal of Combinatorial Theory Series A最新文献

{"title":"Linkage of graphs with flows","authors":"Alex Abreu ,&nbsp;Marco Pacini ,&nbsp;Matheus Secco","doi":"10.1016/j.jcta.2024.105907","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105907","url":null,"abstract":"<div><p>We prove several linkage properties of graphs with flows, generalizing some results on linkage of graphs. This translates in properties of connectedness through codimension one of certain posets. For example, the poset of flows and the posets of odd and even tropical spin curves. These posets are, respectively, the posets underlying the moduli space of roots of divisors on tropical curves and the moduli spaces of odd and even tropical spin curves.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105907"},"PeriodicalIF":1.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140822507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The enumeration of equivalent classes of minimal general dihedral group codes","authors":"Boheng Huang","doi":"10.1016/j.jcta.2024.105906","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105906","url":null,"abstract":"<div><p>A group code is a linear code which can be realized as a two-sided ideal of a group algebra over a finite field. When the characteristic of the field is prime to the order of the group, we will give explicit expressions for primitive central idempotents in the group algebra, which enables us to determine the number of equivalent classes of minimal group codes. Then, we apply our formula to calculate the number of equivalent classes of minimal general dihedral group codes.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105906"},"PeriodicalIF":1.1,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140649718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The number of primitive words of unbounded exponent in the language of an HD0L-system is finite","authors":"Karel Klouda,&nbsp;Štěpán Starosta","doi":"10.1016/j.jcta.2024.105904","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105904","url":null,"abstract":"<div><p>Let <em>H</em> be an HD0L-system. We show that there are only finitely many primitive words <em>v</em> with the property that <span><math><msup><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, for all integers <em>k</em>, is an element of the factorial language of <em>H</em>. In particular, this result applies to the set of all factors of a morphic word. We provide a formalized proof in the proof assistant Isabelle/HOL as part of the Combinatorics on Words Formalized project.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105904"},"PeriodicalIF":1.1,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140633182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Basic tetravalent oriented graphs with cyclic normal quotients","authors":"Nemanja Poznanović ,&nbsp;Cheryl E. Praeger","doi":"10.1016/j.jcta.2024.105895","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105895","url":null,"abstract":"<div><p>Let <span><math><mi>OG</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> denote the family of all graph-group pairs <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span> where Γ is finite, 4-valent, connected, and <em>G</em>-oriented (<em>G</em>-half-arc-transitive). A subfamily of <span><math><mi>OG</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> has recently been identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one basic member. In this paper we provide a description of such basic pairs which have at least one <em>G</em>-normal quotient which is isomorphic to a cycle graph. In doing so, we produce many new infinite families of examples and solve several problems posed in the recent literature on this topic. This result completes a research project aiming to provide a description of all basic pairs in <span><math><mi>OG</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105895"},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000347/pdfft?md5=f4ddc34a4bd5053c5b3e0115d6f19233&pid=1-s2.0-S0097316524000347-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140558761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Explicit formulas for a family of hypermaps beyond the one-face case","authors":"Zi-Wei Bai,&nbsp;Ricky X.F. Chen","doi":"10.1016/j.jcta.2024.105905","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105905","url":null,"abstract":"<div><p>Enumeration of hypermaps (or Grothendieck's dessins d'enfants) is widely studied in many fields. In particular, enumerating hypermaps with a fixed edge-type according to the number of faces and genus is one topic of great interest. The first systematic study of hypermaps with one face and any fixed edge-type is the work of Jackson (1987) <span>[23]</span> using group characters. Stanley later (2011) obtained the genus distribution polynomial of one-face hypermaps of any fixed edge-type expressed in terms of the backward shift operator. There is also enormous amount of work on enumerating one-face hypermaps of specific edge-types. Hypermaps with more faces are generally much harder to enumerate and results are rare. Our main results here are formulas for the genus distribution polynomials for a family of typical two-face hypermaps including almost all edge-types, the purely imaginary zeros property of these polynomials, and the log-concavity of the coefficients.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105905"},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140604786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Proof of a conjecture of Ballantine and Merca on truncated sums of 6-regular partitions","authors":"Olivia X.M. Yao","doi":"10.1016/j.jcta.2024.105903","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105903","url":null,"abstract":"<div><p>In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Their work has opened up a new study of truncated series. Recently, Ballantine and Merca posed a conjecture on infinite families of inequalities involving <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, which counts the number of 6-regular partitions of <em>n</em>. In this paper, we confirm Ballantine and Merca's conjecture on linear equalities of <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> based on a formula of the number of partitions of <em>n</em> into parts not exceeding 3 due to Cayley.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105903"},"PeriodicalIF":1.1,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140554335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Classical groups as flag-transitive automorphism groups of 2-designs with λ = 2","authors":"Seyed Hassan Alavi ,&nbsp;Mohsen Bayat ,&nbsp;Ashraf Daneshkhah ,&nbsp;Marjan Tadbirinia","doi":"10.1016/j.jcta.2024.105892","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105892","url":null,"abstract":"<div><p>In this article, we study 2-designs with <span><math><mi>λ</mi><mo>=</mo><mn>2</mn></math></span> admitting a flag-transitive and point-primitive almost simple automorphism group <em>G</em> with socle <em>X</em> a finite simple classical group of Lie type. We prove that such a design belongs to an infinite family of 2-designs with parameter set <span><math><mo>(</mo><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> and <span><math><mi>X</mi><mo>=</mo><msub><mrow><mi>PSL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mn>3</mn><mo>)</mo></math></span> for some <span><math><mi>n</mi><mo>⩾</mo><mn>3</mn></math></span>, or <span><math><mi>X</mi><mo>=</mo><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> with point-stabiliser <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msub></math></span>, or it is isomorphic to the 2-design with parameter set <span><math><mo>(</mo><mn>6</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>7</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>10</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>11</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>28</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>28</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>36</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> or <span><math><mo>(</mo><mn>126</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105892"},"PeriodicalIF":1.1,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140546056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Further q-reflections on the modulo 9 Kanade–Russell (conjectural) identities","authors":"Stepan Konenkov","doi":"10.1016/j.jcta.2024.105894","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105894","url":null,"abstract":"<div><p>We examine four identities conjectured by Dean Hickerson which complement five modulo 9 Kanade–Russell identities, and we build up a profile of new identities and new conjectures similar to those found by Ali Uncu and Wadim Zudilin.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105894"},"PeriodicalIF":1.1,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140540286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids","authors":"Lilla Tóthmérész","doi":"10.1016/j.jcta.2024.105891","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105891","url":null,"abstract":"<div><p>We define the root polytope of a regular oriented matroid, and show that the greedoid polynomial of an Eulerian branching greedoid rooted at vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is equivalent to the <span><math><msup><mrow><mi>h</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-polynomial of the root polytope of the dual of the graphic matroid.</p><p>As the definition of the root polytope is independent of the vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, this gives a geometric proof for the root-independence of the greedoid polynomial for Eulerian branching greedoids, a fact which was first proved by Swee Hong Chan, Kévin Perrot and Trung Van Pham using sandpile models. We also obtain that the greedoid polynomial does not change if we reverse every edge of an Eulerian digraph.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105891"},"PeriodicalIF":1.1,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009731652400030X/pdfft?md5=24b6ae0df2f0ec32e3f3bfc6f52f2e2c&pid=1-s2.0-S009731652400030X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140347134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cyclic relative difference families with block size four and their applications","authors":"Chenya Zhao ,&nbsp;Binwei Zhao ,&nbsp;Yanxun Chang ,&nbsp;Tao Feng ,&nbsp;Xiaomiao Wang ,&nbsp;Menglong Zhang","doi":"10.1016/j.jcta.2024.105890","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105890","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Given a subgroup &lt;em&gt;H&lt;/em&gt; of a group &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, a &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; difference family (DF) is a set &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of &lt;em&gt;k&lt;/em&gt;-subsets of &lt;em&gt;G&lt;/em&gt; such that &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the subgroup of order &lt;em&gt;h&lt;/em&gt; in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; generated by &lt;em&gt;g&lt;/em&gt;. A &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-DF is called cyclic and written as a &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-CDF. This paper shows that for &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, there exists a &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-CDF if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;⩾&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∉&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. As a corollary, it is shown that a 1-rotational Steiner system S&lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; exists if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. This solves the long-standing open problem on the existence of a 1-rotational S&lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. As another corollary, we establish the existence of an optimal &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-optical orthogonal code with &lt;span&gt;&lt;math&gt;&lt;mo&gt;⌊&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;−&lt;/","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105890"},"PeriodicalIF":1.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140342255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}