Journal of Combinatorial Designs最新文献

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On large partial ovoids of symplectic and Hermitian polar spaces 关于辛和Hermitian极空间的大偏卵形
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-11-06 DOI: 10.1002/jcd.21864
Michela Ceria, Jan De Beule, Francesco Pavese, Valentino Smaldore
{"title":"On large partial ovoids of symplectic and Hermitian polar spaces","authors":"Michela Ceria, Jan De Beule, Francesco Pavese, Valentino Smaldore","doi":"10.1002/jcd.21864","DOIUrl":"https://doi.org/10.1002/jcd.21864","url":null,"abstract":"<p>In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>W</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${mathscr{W}}(3,q)$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math> odd square, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 \u0000 <mo>≢</mo>\u0000 \u0000 <mn>0</mn>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $qnotequiv 0(mathrm{mod}3)$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>W</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>5</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${mathscr{W}}(5,q)$</annotation>\u0000 </semantics></math> and of the Hermitian polar spaces <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℋ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal H} }}(4,{q}^{2})$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 <","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 1","pages":"5-22"},"PeriodicalIF":0.7,"publicationDate":"2022-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50123308","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
An alternative construction of the Hermitian unital 2-(28, 4, 1) design Hermitian酉2-(28, 4. 1) 设计
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-09-26 DOI: 10.1002/jcd.21861
Koichi Inoue
{"title":"An alternative construction of the Hermitian unital 2-(28, 4, 1) design","authors":"Koichi Inoue","doi":"10.1002/jcd.21861","DOIUrl":"https://doi.org/10.1002/jcd.21861","url":null,"abstract":"<p>In this paper, we give an alternative construction of the Hermitian unital 2-(28, 4, 1) design in such a way that it is constructed on the isotropic vectors in a unitary geometry of dimension 3 over the field <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{4}$</annotation>\u0000 </semantics></math>. As a corollary, we can construct a unique 3-(10, 4, 1) design (namely, the Witt system <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>W</mi>\u0000 \u0000 <mn>10</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{boldsymbol{W}}}_{{bf{10}}}$</annotation>\u0000 </semantics></math>).</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 12","pages":"752-759"},"PeriodicalIF":0.7,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72166349","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
An alternative construction of the Hermitian unital 2‐(28, 4, 1) design 厄米单位2‐(28,4,1)设计的另一种结构
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-09-26 DOI: 10.1002/jcd.21861
Koichi Inoue
{"title":"An alternative construction of the Hermitian unital 2‐(28, 4, 1) design","authors":"Koichi Inoue","doi":"10.1002/jcd.21861","DOIUrl":"https://doi.org/10.1002/jcd.21861","url":null,"abstract":"In this paper, we give an alternative construction of the Hermitian unital 2‐(28, 4, 1) design in such a way that it is constructed on the isotropic vectors in a unitary geometry of dimension 3 over the field F 4 ${{mathbb{F}}}_{4}$ . As a corollary, we can construct a unique 3‐(10, 4, 1) design (namely, the Witt system W 10 ${{boldsymbol{W}}}_{{bf{10}}}$ ).","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"6 1","pages":"753 - 759"},"PeriodicalIF":0.7,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86699161","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
Weak sequenceability in cyclic groups 循环群中的弱序列性
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-09-26 DOI: 10.1002/jcd.21862
Simone Costa, Stefano Della Fiore
{"title":"Weak sequenceability in cyclic groups","authors":"Simone Costa,&nbsp;Stefano Della Fiore","doi":"10.1002/jcd.21862","DOIUrl":"https://doi.org/10.1002/jcd.21862","url":null,"abstract":"&lt;p&gt;A subset &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $A$&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 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is &lt;i&gt;sequenceable&lt;/i&gt; if there is an ordering &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;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&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;/mrow&gt;\u0000 &lt;annotation&gt; $({a}_{1},ldots ,{a}_{k})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of its elements such that the partial sums &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;msub&gt;\u0000 &lt;mi&gt;s&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;mi&gt;s&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 12","pages":"735-751"},"PeriodicalIF":0.7,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21862","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72166654","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: 4
Completing the spectrum of semiframes with block size three 完成块大小为3的半帧频谱
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-09-13 DOI: 10.1002/jcd.21856
H. Cao, D. Xu, Hao Zheng
{"title":"Completing the spectrum of semiframes with block size three","authors":"H. Cao, D. Xu, Hao Zheng","doi":"10.1002/jcd.21856","DOIUrl":"https://doi.org/10.1002/jcd.21856","url":null,"abstract":"A k ‐semiframe of type g u is a k ‐GDD of type g u ( X , G , ℬ ) , in which the collection of blocks ℬ can be written as a disjoint union ℬ = P ∪ Q , where P is partitioned into parallel classes of X and Q is partitioned into holey parallel classes, each holey parallel class being a partition of X G for some G ∈ G . In this paper, we will introduce a new concept of t ‐perfect semiframe and use it to prove the existence of a 3‐semiframe of type g u with even group size. This completes the proof of the existence of 3‐semiframes with uniform group size.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"5 1","pages":"716 - 732"},"PeriodicalIF":0.7,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76964572","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: 1
Completing the spectrum of semiframes with block size three 完成块大小为3的半帧的谱
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-09-13 DOI: 10.1002/jcd.21856
H. Cao, D. Xu, H. Zheng
{"title":"Completing the spectrum of semiframes with block size three","authors":"H. Cao,&nbsp;D. Xu,&nbsp;H. Zheng","doi":"10.1002/jcd.21856","DOIUrl":"https://doi.org/10.1002/jcd.21856","url":null,"abstract":"<p>A <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-semiframe of type <math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>g</mi>\u0000 \u0000 <mi>u</mi>\u0000 </msup>\u0000 </mrow></math> is a <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-GDD of type <math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>g</mi>\u0000 \u0000 <mi>u</mi>\u0000 </msup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ℬ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math>, in which the collection of blocks <math>\u0000 \u0000 <mrow>\u0000 <mi>ℬ</mi>\u0000 </mrow></math> can be written as a disjoint union <math>\u0000 \u0000 <mrow>\u0000 <mi>ℬ</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>P</mi>\u0000 \u0000 <mo>∪</mo>\u0000 \u0000 <mi>Q</mi>\u0000 </mrow></math>, where <math>\u0000 \u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow></math> is partitioned into parallel classes of <math>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow></math> and <math>\u0000 \u0000 <mrow>\u0000 <mi>Q</mi>\u0000 </mrow></math> is partitioned into holey parallel classes, each holey parallel class being a partition of <math>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo></mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow></math> for some <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow></math>. In this paper, we will introduce a new concept of <math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow></math>-perfect semiframe and use it to prove the existence of a 3-semiframe of type <math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>g</mi>\u0000 \u0000 <mi>u</mi>\u0000 </msup>\u0000 </mrow></math> with even group size. This completes the proof of the existence of 3-semiframes with uniform group size.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 11","pages":"716-732"},"PeriodicalIF":0.7,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72169121","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: 1
Stability of Erdős–Ko–Rado theorems in circle geometries 圆几何中Erdõs–Ko–Rado定理的稳定性
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-08-13 DOI: 10.1002/jcd.21854
Sam Adriaensen
{"title":"Stability of Erdős–Ko–Rado theorems in circle geometries","authors":"Sam Adriaensen","doi":"10.1002/jcd.21854","DOIUrl":"https://doi.org/10.1002/jcd.21854","url":null,"abstract":"<p>Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite nonovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}$</annotation>\u0000 </semantics></math> in one of the known finite circle geometries of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>, with <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>ℱ</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mfrac>\u0000 <mn>1</mn>\u0000 \u0000 <msqrt>\u0000 <mn>2</mn>\u0000 </msqrt>\u0000 </mfrac>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <msqrt>\u0000 <mn>2</mn>\u0000 </msqrt>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>8</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $| {rm{ {mathcal F} }},| ge frac{1}{sqrt{2}}{q}^{2}+2sqrt{2}q+8$</annotation>\u0000 </semantics></math>, must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 11","pages":"689-715"},"PeriodicalIF":0.7,"publicationDate":"2022-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72149789","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: 1
On flag‐transitive 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ designs with λ∣k $lambda | k$ 基于λ∣k $lambda | k$的标志传递2‐(k2,k,λ) $({k}^{2},k,lambda)$设计
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-07-29 DOI: 10.1002/jcd.21852
Alessandro Montinaro, Eliana Francot
{"title":"On flag‐transitive 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ designs with λ∣k $lambda | k$","authors":"Alessandro Montinaro, Eliana Francot","doi":"10.1002/jcd.21852","DOIUrl":"https://doi.org/10.1002/jcd.21852","url":null,"abstract":"It is shown that, apart from the smallest Ree group, a flag‐transitive automorphism group G $G$ of a 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ design D ${mathscr{D}}$ , with λ∣k $lambda | k$ , is either an affine group or an almost simple classical group. Moreover, when G $G$ is the smallest Ree group, D ${mathscr{D}}$ is isomorphic either to the 2‐ (62,6,2) $({6}^{2},6,2)$ design or to one of the three 2‐ (62,6,6) $({6}^{2},6,6)$ designs constructed in this paper. All the four 2‐designs have the 36 secants of a non‐degenerate conic C ${mathscr{C}}$ of PG2(8) $P{G}_{2}(8)$ as a point set and 6‐sets of secants in a remarkable configuration as a block set.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"20 1","pages":"653 - 670"},"PeriodicalIF":0.7,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76703413","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: 2
On flag-transitive 2- ( k 2 , k , λ ) $({k}^{2},k,lambda )$ designs with λ ∣ k $lambda | k$ 关于λ为的标志传递2-(k2,k,λ)$({k}^{2},k,lambda)$设计k$lambda|k$
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-07-29 DOI: 10.1002/jcd.21852
Alessandro Montinaro, Eliana Francot
{"title":"On flag-transitive 2-\u0000 \u0000 \u0000 (\u0000 \u0000 \u0000 k\u0000 2\u0000 \u0000 ,\u0000 k\u0000 ,\u0000 λ\u0000 \u0000 )\u0000 \u0000 $({k}^{2},k,lambda )$\u0000 designs with \u0000 \u0000 \u0000 λ\u0000 ∣\u0000 k\u0000 \u0000 $lambda | k$","authors":"Alessandro Montinaro,&nbsp;Eliana Francot","doi":"10.1002/jcd.21852","DOIUrl":"https://doi.org/10.1002/jcd.21852","url":null,"abstract":"&lt;p&gt;It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group &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; of a 2-&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&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;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $({k}^{2},k,lambda )$&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; ${mathscr{D}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, with &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;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $lambda | k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, is either an affine group or an almost simple classical group. Moreover, when &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; is the smallest Ree group, &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; ${mathscr{D}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is isomorphic either to the 2-&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $({6}^{2},6,2)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; design or to one of the three 2-&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;6&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $({6}^{2},6,6)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; designs constructed in this paper. All the four 2-designs have the 36 secants of a non-degenerate conic &lt;math&gt;\u0000 &lt;sem","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 10","pages":"653-670"},"PeriodicalIF":0.7,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72192571","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: 2
The spectrum for large sets of resolvable idempotent Latin squares 大集合可分解幂等拉丁平方的谱
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2022-07-27 DOI: 10.1002/jcd.21853
Xiangqian Li, Yanxun Chang
{"title":"The spectrum for large sets of resolvable idempotent Latin squares","authors":"Xiangqian Li,&nbsp;Yanxun Chang","doi":"10.1002/jcd.21853","DOIUrl":"https://doi.org/10.1002/jcd.21853","url":null,"abstract":"&lt;p&gt;An idempotent Latin square of order &lt;math&gt;\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 called resolvable and denoted by RILS(&lt;i&gt;v&lt;/i&gt;) if the &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\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 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v(v-1)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; off-diagonal cells can be resolved into &lt;math&gt;\u0000 &lt;semantics&gt;\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 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v-1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; disjoint transversals. A large set of resolvable idempotent Latin squares of order &lt;math&gt;\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;, briefly LRILS(&lt;i&gt;v&lt;/i&gt;), is a collection of &lt;math&gt;\u0000 &lt;semantics&gt;\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;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v-2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; RILS(&lt;i&gt;v&lt;/i&gt;)s pairwise agreeing on only the main diagonal. In this article, an LRILS(&lt;i&gt;v&lt;/i&gt;) is constructed for &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&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;14&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;20&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;22&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;28&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;34&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;35&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;38&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;40&lt;/mn&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 10","pages":"671-683"},"PeriodicalIF":0.7,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72166526","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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