Journal of Combinatorial Designs最新文献_第5页

Dual incidences and t-designs in vector spaces 向量空间中的对偶关联和t-设计

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-10-13 DOI: 10.1002/jcd.21922

Kristijan Tabak

Let <math> <semantics> <mrow> <mi>V</mi> </mrow> <annotation> $V$</annotation> </semantics></math> be an <math> <semantics> <mrow> <mi>n</mi> </mrow> <annotation> $n$</annotation> </semantics></math>-dimensional vector space over <math> <semantics> <mrow> <msub> <mi>F</mi> <mi>q</mi> </msub> </mrow> <annotation> ${{mathbb{F}}}_{q}$</annotation> </semantics></math> and <math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> ${rm{ {mathcal H} }}$</annotation> </semantics></math> is any set of <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-dimensional subspaces of <math> <semantics> <mrow> <mi>V</mi> </mrow> <annotation> $V$</annotation> </semantics></math>. We construct two incidence structures <math> <semantics> <mrow> <msub> <mi>D</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>H</mi> <mo>)</mo> </mrow> </mrow> <annotation> ${{mathscr{D}}}_{max}({rm{ {mathcal H} }})$</annotation> </semantics></math> and <math> <semantics> <mrow> <msub> <mi>D</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>H</mi> <mo>)</mo> </mrow> </mrow> <annotation> ${{mathscr{D}}}_{min}({rm{ {mathcal H} }})$</annotation> </semantics></math> using subspaces from <math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> ${rm{ {mathcal H} }}$</annotation>

设V$ V$是F $ q ${{mathbb{F}}}_{q}$和H上的n$ n维向量空间${rm{{mathcal H}}}$是V$ V$的k$ k$维子空间的任意集合。我们构造了两个关联结构D m a x (H)${{mathscr{D}}}_{max}({rm{{mathcal H}}})$和D m i n (H)${{mathscr{D}}}_{min}({rm{mathcal H}}})$使用H ${rm{{mathcal H}}}} $的子空间。这些点是H ${rm{{mathcal H}}}$的子空间。dmma x (H)的块${{mathscr{D}}}_{max}({rm{{mathcal H}}})$由V$ V$的所有超平面索引，而dm块i n (H)${{mathscr{D}}}_{min}({rm{{mathcal H}}})$由维度为1的所有子空间索引。我们证明了dmma x (H)${{mathscr{D}}}_{max}({rm{{mathcal H}}})$和D m i n (H)${{mathscr{D}}}_{min}({rm{{mathcal H}}})$是对偶的，因为它们的关联矩阵是相关的，一个可以从另一个计算出来。另外，如果H ${rm{{mathcal H}}}$是t−(n)，k ,λ) q $t-{(n,k， λ)}_{q}$ -设计证明了D m关联矩阵的新矩阵方程ax (H)$ {{mathscr{D}}}_{max}({rm{mathcal H}}})$和Dmin (H)$ {{mathscr{D}}}_{min}({rm{{mathcal H}}})$。

{"title":"Dual incidences and t-designs in vector spaces","authors":"Kristijan Tabak","doi":"10.1002/jcd.21922","DOIUrl":"https://doi.org/10.1002/jcd.21922","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 </mrow>\u0000 <annotation> $V$</annotation>\u0000 </semantics></math> be an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-dimensional vector space over <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{q}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal H} }}$</annotation>\u0000 </semantics></math> is any set of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-dimensional subspaces of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 </mrow>\u0000 <annotation> $V$</annotation>\u0000 </semantics></math>. We construct two incidence structures <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>D</mi>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mi>a</mi>\u0000 \u0000 <mi>x</mi>\u0000 </mrow>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{mathscr{D}}}_{max}({rm{ {mathcal H} }})$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>D</mi>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mi>i</mi>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{mathscr{D}}}_{min}({rm{ {mathcal H} }})$</annotation>\u0000 </semantics></math> using subspaces from <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal H} }}$</annotation>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 1","pages":"46-52"},"PeriodicalIF":0.7,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134805714","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

Completing the solution of the directed Oberwolfach problem with cycles of equal length 完成了等长环有向Oberwolfach问题的求解

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-09-25 DOI: 10.1002/jcd.21918

Alice Lacaze-Masmonteil

In this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two-table case with tables of equal odd length. We prove that the complete symmetric digraph on $� � � 2 � m � � �$ vertices, denoted $� � � �_{K}^{�} � � �$ , admits a resolvable decomposition into directed cycles of odd length $� � � m � � �$ . This completely settles the directed Oberwolfach problem with tables of uniform length.

本文给出了具有等长表的有向Oberwolfach问题的最后一个突出情况的解。也就是说，我们用奇数长度相等的表来处理双表的情况。我们证明了2m$ 2m$顶点上的完全对称有向图，记作K 2m * ${K}_{2m}^{*}$，允许分解成奇数长度m$ m$的有向环。这完全解决了等长表的有向奥伯沃尔法赫问题。

{"title":"Completing the solution of the directed Oberwolfach problem with cycles of equal length","authors":"Alice Lacaze-Masmonteil","doi":"10.1002/jcd.21918","DOIUrl":"https://doi.org/10.1002/jcd.21918","url":null,"abstract":"In this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two-table case with tables of equal odd length. We prove that the complete symmetric digraph on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $2m$</annotation>\u0000 </semantics></math> vertices, denoted <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>K</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <mo>*</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation> ${K}_{2m}^{* }$</annotation>\u0000 </semantics></math>, admits a resolvable decomposition into directed cycles of odd length <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>. This completely settles the directed Oberwolfach problem with tables of uniform length.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 1","pages":"5-30"},"PeriodicalIF":0.7,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21918","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134880386","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: 3

Orthogonal cycle systems with cycle length less than 10 周期长度小于10的正交循环系

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-09-25 DOI: 10.1002/jcd.21921

Selda Küçükçifçi, Emine Şule Yazıcı

An <math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> $H$</annotation> </semantics></math>-decomposition of a graph <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> is a partition of the edge set of <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> into subsets, where each subset induces a copy of the graph <math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> $H$</annotation> </semantics></math>. A <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-orthogonal <math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> $H$</annotation> </semantics></math>-decomposition of <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> is a set of <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math><math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> $H$</annotation> </semantics></math>-decompositions of <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> such that any two copies of <math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> $H$</annotation> </semantics></math> in distinct <math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> $H$</annotation> </semantics></math>-decompositions intersect in at most one edge. When <math> <semantics> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mi>K</mi> <mi>v</mi> </msub> </mrow> <annotation> $G={K}_{v}$</annotation> </semantics></math>, we call the <math> <semantics> <mrow> <mi>H</mi> </mrow> <annotation> $H$</annotation> </semantics></math>-decompositi

图G$ G$的H$ H$分解是将G$ G$的边集划分为若干子集，其中每个子集归纳出图H$ H$的一个副本。G$ G$的k$ k$ -正交H$ H$分解是k$ k$ H$ H$的集合- G$ G$的分解使得H$ H$的任意两个副本在不同的H$ H$分解中最多相交于一条边。当G= K v $G={K}_{v}$时，我们称之为H$ H$分解和H$ H$ - v$ v$阶系统。在本文中，我们考虑了H$ H$是一个r $ell $ -环的情况，并构造了一对r $ell $ -环的正交系统{5,6,7,8,9}$ well in {5,6,7,8,9}$，除了当r =v$ r =v$。

{"title":"Orthogonal cycle systems with cycle length less than 10","authors":"Selda Küçükçifçi, Emine Şule Yazıcı","doi":"10.1002/jcd.21921","DOIUrl":"https://doi.org/10.1002/jcd.21921","url":null,"abstract":"An <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>-decomposition of a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a partition of the edge set of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> into subsets, where each subset induces a copy of the graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>. A <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-orthogonal <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>-decomposition of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a set of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>-decompositions of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that any two copies of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> in distinct <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>-decompositions intersect in at most one edge. When <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>K</mi>\u0000 <mi>v</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $G={K}_{v}$</annotation>\u0000 </semantics></math>, we call the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>-decompositi","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 1","pages":"31-45"},"PeriodicalIF":0.7,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134814340","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

Generalizations of some Nordhaus–Gaddum-type results on spectral radius 谱半径上某些Nordhaus–Gaddum型结果的推广

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-09-21 DOI: 10.1002/jcd.21919

Junying Lu, Lanchao Wang, Yaojun Chen

Let <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> be a simple graph and <math> <semantics> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> <annotation> $lambda (G)$</annotation> </semantics></math> the spectral radius of <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math>. For <math> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> <annotation> $kge 2$</annotation> </semantics></math>, a <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-edge decomposition <math> <semantics> <mrow> <mo>(</mo> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> <annotation> $({H}_{1},{rm{ldots }},{H}_{k})$</annotation> </semantics></math> is <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math> spanning subgraphs such that their edge sets form a <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-partition of the edge set of <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math>. In this paper, we obtain some sharp lower and upper bounds for <math> <semantics> <mrow> <mi>λ</mi> <mrow> <mo>(</mo>

设G$G$是一个简单图，λ（G）$lambda（G）$是谱G$G$的半径。对于k≥2$kge 2$，k$k$边分解（h1，…，Hk）$({H}_{1} ，｛rm｛ldots｝｝，{H}_{k} ）$是k$k$生成子图，使得它们的边集形成G$G$的边集的k$k$-划分。在本文中，我们得到了λ（H1）+…+λ（H k）$lambda({H}_{1} ）+，cdots，+lambda({H}_{k} ）$根据H i的集团数${H}_{i} $和G$G$的大小，并讨论什么是k$k$-边分解（h1，…，Hk）$({H}_{1} ，｛rm｛ldots｝｝，{H}_{k} ）$可以最大化λ（H1）+…+λ（H k）$lambda({H}_{1} ）+cdots，+lambda({H}_{k} ）$，当G$G$是一个完全图时。这些结果推广了Nosal、Hong和Shu以及Nikiforov关于k=2$k=2$的谱半径的一些Nordhaus–Gaddum型结果。

{"title":"Generalizations of some Nordhaus–Gaddum-type results on spectral radius","authors":"Junying Lu, Lanchao Wang, Yaojun Chen","doi":"10.1002/jcd.21919","DOIUrl":"https://doi.org/10.1002/jcd.21919","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> be a simple graph and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $lambda (G)$</annotation>\u0000 </semantics></math> the spectral radius of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. For <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $kge 2$</annotation>\u0000 </semantics></math>, a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-edge decomposition <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>H</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $({H}_{1},{rm{ldots }},{H}_{k})$</annotation>\u0000 </semantics></math> is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> spanning subgraphs such that their edge sets form a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-partition of the edge set of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. In this paper, we obtain some sharp lower and upper bounds for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 12","pages":"701-712"},"PeriodicalIF":0.7,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50153173","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

Construction of rank 4 self-dual association schemes inducing three partial geometric designs 诱导三个局部几何设计的秩为4的自对偶关联方案的构造

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-09-05 DOI: 10.1002/jcd.21917

Akihide Hanaki

Xu characterized rank 4 self-dual association schemes inducing three partial geometric designs by their character tables. We construct such association schemes as Schur rings over Abelian 2-groups.

徐用秩4自对偶关联方案的特征表刻画了它们的三个局部几何设计。我们构造了阿贝尔2-群上的Schur环这样的关联方案。

引用次数: 0

proper partial geometries with an automorphism group acting primitively on points and lines 自同构群初始作用于点和线的适当局部几何

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-08-31 DOI: 10.1002/jcd.21914

Wendi Di

Let $� � � S � � �$ be a finite proper partial geometry pg $� � � � (� � s � �, � � t � �, � � α � � �) � � � �$ not isomorphic to the van Lint–Schrijver partial geometry pg $� � � � (� � 5 � �, � � 5 � �, � � 2 � � �) � � � �$ and let $� � � G � � �$ be a group of automorphisms of $� � � S � � �$ acting primitively on both points and lines of $� � � S � � �$ , we show that if $� � � α � � \leq � � 60 � � �$ then $� � � G � � �$ must be almost simple.

设S$｛mathscr｛S｝｝$是一个有限的正偏几何pg（s，t，α）$（s，talpha）$不同构到van Lint–Schrijver局部几何pg（5，5，2）$（5,5,2）$并出租G$G$是S$｛mathscr｛S｝｝$的一组自同构，我们证明了如果α≤60$alphale60$，那么G$G$必须是几乎简单的。

{"title":"proper partial geometries with an automorphism group acting primitively on points and lines","authors":"Wendi Di","doi":"10.1002/jcd.21914","DOIUrl":"https://doi.org/10.1002/jcd.21914","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math> be a finite proper partial geometry pg<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>t</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>α</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(s,t,alpha )$</annotation>\u0000 </semantics></math> not isomorphic to the van Lint–Schrijver partial geometry pg<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>5</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>5</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(5,5,2)$</annotation>\u0000 </semantics></math> and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> be a group of automorphisms of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math> acting primitively on both points and lines of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math>, we show that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>60</mn>\u0000 </mrow>\u0000 <annotation> $alpha le 60$</annotation>\u0000 </semantics></math> then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> must be almost simple.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"642-664"},"PeriodicalIF":0.7,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50148812","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

P℘N functions, complete mappings and quasigroup difference sets P℘N函数、完备映射与拟群差集

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-08-23 DOI: 10.1002/jcd.21916

Nurdagül Anbar, Tekgül Kalaycı, Wilfried Meidl, Constanza Riera, Pantelimon Stănică

We investigate pairs of permutations <math> <semantics> <mrow> <mi>F</mi> <mo>,</mo> <mi>G</mi> </mrow> <annotation> $F,G$</annotation> </semantics></math> of <math> <semantics> <mrow> <msub> <mi>F</mi> <msup> <mi>p</mi> <mi>n</mi> </msup> </msub> </mrow> <annotation> ${{mathbb{F}}}_{{p}^{n}}$</annotation> </semantics></math> such that <math> <semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <annotation> $F(x+a)-G(x)$</annotation> </semantics></math> is a permutation for every <math> <semantics> <mrow> <mi>a</mi> <mo>∈</mo> <msub> <mi>F</mi> <msup> <mi>p</mi> <mi>n</mi> </msup> </msub> </mrow> <annotation> $ain {{mathbb{F}}}_{{p}^{n}}$</annotation> </semantics></math>. We show that, in that case, necessarily <math> <semantics> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>℘</mi> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo>

我们研究了排列对F、G$F，Fp n$的G$｛mathbb｛F｝｝｝_｛p｝^｛n｝｝$使得F（x+a）−G（x）$F（x+a）-G（x）$是每个a∈F p的置换n$ain｛mathbb｛F｝｝｝_｛p｝^｛n｝｝$。我们证明，在这种情况下，G（x）=℘ （F（x））$G（x）=wp（F（x））$对于一些完整的映射−℘ $-｛mathbb｛F｝｝_｛p｝^｛n｝$的wp$，并称置换F$F$为完美℘ $wp$非线性（P℘ $wp$N）函数。如果℘ 则F$F$是PcN函数，这在最近的文献中被考虑过。用F上的二进制运算p n$｛mathbb｛F｝｝｝_℘ $wp$，我们得到了一个拟群，并证明了一个P的图℘ $wp$N函数F$F$是在相应的拟群中的差集。我们进一步指出了从这种拟群差集获得的对称设计的变体。最后，我们分析了P的等价性（通过相应拟群的自同构群自然定义）℘ $wp$N函数。

{"title":"P℘N functions, complete mappings and quasigroup difference sets","authors":"Nurdagül Anbar, Tekgül Kalaycı, Wilfried Meidl, Constanza Riera, Pantelimon Stănică","doi":"10.1002/jcd.21916","DOIUrl":"https://doi.org/10.1002/jcd.21916","url":null,"abstract":"We investigate pairs of permutations <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $F,G$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <msup>\u0000 <mi>p</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msup>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{{p}^{n}}$</annotation>\u0000 </semantics></math> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>a</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>G</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>x</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $F(x+a)-G(x)$</annotation>\u0000 </semantics></math> is a permutation for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <msup>\u0000 <mi>p</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msup>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $ain {{mathbb{F}}}_{{p}^{n}}$</annotation>\u0000 </semantics></math>. We show that, in that case, necessarily <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>x</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>℘</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>x</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 12","pages":"667-690"},"PeriodicalIF":0.7,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50141470","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 the directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform-length cycles 关于完全对称等分有向图和一致长环的有向Oberwolfach问题

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-08-22 DOI: 10.1002/jcd.21913

Nevena Francetić, Mateja Šajna

We examine the necessary and sufficient conditions for a complete symmetric equipartite digraph <math> <semantics> <mrow> <msubsup> <mi>K</mi> <mrow> <mi>n</mi> <mrow> <mo>[</mo> <mi>m</mi> <mo>]</mo> </mrow> </mrow> <mo>*</mo> </msubsup> </mrow> <annotation> ${K}_{n[m]}^{* }$</annotation> </semantics></math> with <math> <semantics> <mrow> <mi>n</mi> </mrow> <annotation> $n$</annotation> </semantics></math> parts of size <math> <semantics> <mrow> <mi>m</mi> </mrow> <annotation> $m$</annotation> </semantics></math> to admit a resolvable decomposition into directed cycles of length <math> <semantics> <mrow> <mi>t</mi> </mrow> <annotation> $t$</annotation> </semantics></math>. We show that the obvious necessary conditions are sufficient for <math> <semantics> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>t</mi> <mo>≥</mo> <mn>2</mn> </mrow> <annotation> $m,n,tge 2$</annotation> </semantics></math> in each of the following four cases: (i) <math> <semantics> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $m(n-1)$</annotation> </semantics></math> is even; (ii) <math> <semantics> <mrow> <mtext>gcd</mtext> <mrow> <mo>

我们检验了完全对称等分有向图Kn的充要条件[m]*${K}_{n[m]}^{*}$，具有m$m$大小的n$n$部分，以允许将可分解分解为长度为t的有向循环$t$。我们证明了m，t≥2$m，在以下四种情况下各支付2美元：（i）m（n−1）$m（n-1）$是偶数；（ii）gcd（m，n）∉｛1，3｝$text｛gcd｝（m，n）notin｛1,3｝$；（iii）gcd（m，n）=1$text｛gcd｝（m，n）=1$和4Şn$4|n$或6|n$6|n$；和（iv）gcd（m，n）=3$text｛gcd｝（m，n）=3$，并且如果n=6$n=6$，则素数p≤37的p|m$p|m$37美元。

{"title":"On the directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform-length cycles","authors":"Nevena Francetić, Mateja Šajna","doi":"10.1002/jcd.21913","DOIUrl":"https://doi.org/10.1002/jcd.21913","url":null,"abstract":"We examine the necessary and sufficient conditions for a complete symmetric equipartite digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>K</mi>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>*</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation> ${K}_{n[m]}^{* }$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> parts of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> to admit a resolvable decomposition into directed cycles of length <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>. We show that the obvious necessary conditions are sufficient for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>t</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $m,n,tge 2$</annotation>\u0000 </semantics></math> in each of the following four cases: (i) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m(n-1)$</annotation>\u0000 </semantics></math> is even; (ii) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>gcd</mtext>\u0000 \u0000 <mrow>\u0000 <mo>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"604-641"},"PeriodicalIF":0.7,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21913","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50140497","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: 0

A generalization of group divisible t $t$ -designs 群可整除t$t$-设计的一个推广

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-08-10 DOI: 10.1002/jcd.21912

Sijia Liu, Yue Han, Lijun Ma, Lidong Wang, Zihong Tian

Cameron defined the concept of generalized $� � � t � � �$ -designs, which generalized $� � � t � � �$ -designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized $� � � t � � �$ -designs and group divisible $� � � t � � �$ -designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible $� � � t � � �$ -designs, and then point out close connections with various well-known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible $� � � t � � �$ -designs and almost completely determine their existence for $� � � t � = � 2 �, � 3 � � �$ and small block sizes.

Cameron定义了广义t$t$-设计的概念，包括广义t$t$-设计、可分解设计和正交阵列。本文介绍了一类新的组合设计，它同时提供了广义t$t$-设计和群可分t$t$-设计的推广。在某些情况下，我们导出了广义群可整除t$t$-设计存在的必要条件，然后指出了与各种众所周知的设计类的密切联系，包括混合正交阵、完全多部分图的因子分解，大集合的群可分割设计和具有（正交）可分解性的群可划分设计。此外，我们研究了广义群可整除t$t$-设计的构造，并几乎完全确定了它们在t=2时的存在性，3$t=2,3$和小块大小。

{"title":"A generalization of group divisible \u0000 \u0000 \u0000 t\u0000 \u0000 $t$\u0000 -designs","authors":"Sijia Liu, Yue Han, Lijun Ma, Lidong Wang, Zihong Tian","doi":"10.1002/jcd.21912","DOIUrl":"https://doi.org/10.1002/jcd.21912","url":null,"abstract":"Cameron defined the concept of generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, which generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs and group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, and then point out close connections with various well-known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs and almost completely determine their existence for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation> $t=2,3$</annotation>\u0000 </semantics></math> and small block sizes.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"575-603"},"PeriodicalIF":0.7,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50137077","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

Cameron–Liebler sets for maximal totally isotropic flats in classical affine spaces 经典仿射空间中最大全各向同性平面的Cameron–Liebler集

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-07-19 DOI: 10.1002/jcd.21909

Jun Guo, Lingyu Wan

Let <math> <semantics> <mrow> <mi>A</mi> <mi>C</mi> <mi>G</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>ν</mi> <mo>,</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $ACG(2nu ,{{mathbb{F}}}_{q})$</annotation> </semantics></math> be the <math> <semantics> <mrow> <mn>2</mn> <mi>ν</mi> </mrow> <annotation> $2nu $</annotation> </semantics></math>-dimensional classical affine space with parameter <math> <semantics> <mrow> <mi>e</mi> </mrow> <annotation> $e$</annotation> </semantics></math> over a <math> <semantics> <mrow> <mi>q</mi> </mrow> <annotation> $q$</annotation> </semantics></math>-element finite field <math> <semantics> <mrow> <msub> <mi>F</mi> <mi>q</mi> </msub> </mrow> <annotation> ${{mathbb{F}}}_{q}$</annotation> </semantics></math>, and <math> <semantics> <mrow> <msub> <mi>O</mi> <mi>ν</mi> </msub> </mrow> <annotation> ${{mathscr{O}}}_{nu }$</annotation> </semantics></math> be the set of all maximal totally isotropic flats in <math> <semantics> <mrow> <mi>A</mi> <mi>C</mi> <mi>G</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>ν</mi> <mo>,</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $ACG(2nu ,{{mathbb{F}}}_{q})$</annotation> </semantics></math>. In this paper, we discuss Cameron–Liebler sets in <math> <semantics> <mrow> <msub> <mi>O</mi> <mi>ν</mi> </msub> </mrow> <annotation> ${{mathscr{O}}}_{nu }$</annotation> </semantics></math>

设A C G（2Γ，Fq）$ACG（2nu，｛mathbb｛F｝｝}_｛q｝）$是q上参数为e$e$的2μ$2nu$维经典仿射空间$q$元有限域Fq$｛mathbb｛F｝｝_，并且OΓ$｛mathscr｛O｝｝｝_｛nu｝$是A C G中所有最大全各向同性平面的集合（2Γ，Fq）$ACG（2nu，｛mathbb｛F｝｝}_｛q｝）$。本文讨论了OΓ$｛mathscr｛O｝｝_｛nu｝$中的Cameron–Liebler集，得到了几个等价的定义，并给出了一些分类结果。

{"title":"Cameron–Liebler sets for maximal totally isotropic flats in classical affine spaces","authors":"Jun Guo, Lingyu Wan","doi":"10.1002/jcd.21909","DOIUrl":"https://doi.org/10.1002/jcd.21909","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>C</mi>\u0000 <mi>G</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>ν</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $ACG(2nu ,{{mathbb{F}}}_{q})$</annotation>\u0000 </semantics></math> be the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>ν</mi>\u0000 </mrow>\u0000 <annotation> $2nu $</annotation>\u0000 </semantics></math>-dimensional classical affine space with parameter <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow>\u0000 <annotation> $e$</annotation>\u0000 </semantics></math> over a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>-element finite field <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{q}$</annotation>\u0000 </semantics></math>, and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>O</mi>\u0000 <mi>ν</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathscr{O}}}_{nu }$</annotation>\u0000 </semantics></math> be the set of all maximal totally isotropic flats in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>C</mi>\u0000 <mi>G</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>ν</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $ACG(2nu ,{{mathbb{F}}}_{q})$</annotation>\u0000 </semantics></math>. In this paper, we discuss Cameron–Liebler sets in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>O</mi>\u0000 <mi>ν</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathscr{O}}}_{nu }$</annotation>\u0000 </semantics></math>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"547-574"},"PeriodicalIF":0.7,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50137650","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