Journal of Combinatorial Designs最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}