Journal of Combinatorial Designs最新文献

英文中文

Existence of small ordered orthogonal arrays 小有序正交阵列的存在性

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-06-19 DOI: 10.1002/jcd.21903

Kai-Uwe Schmidt, Charlene Weiß

We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.

我们证明了存在有序正交阵列，其大小偏离Rao界的因素是有序正交阵列参数中的多项式。该证明是非结构化的，基于Kuperberg、Lovett和Peled的概率方法。

引用次数: 0

Linear and circular single-change covering designs revisited 重新审视线性和圆形单次变更覆盖设计

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-06-01 DOI: 10.1002/jcd.21885

Amanda Chafee, Brett Stevens

A single-change covering design (SCCD) is a <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math>-set <math> <semantics> <mrow> <mi>X</mi> </mrow> <annotation> $X$</annotation> </semantics></math> and an ordered list <math> <semantics> <mrow> <mi>ℒ</mi> </mrow> <annotation> ${rm{ {mathcal L} }}$</annotation> </semantics></math> of <math> <semantics> <mrow> <mi>b</mi> </mrow> <annotation> $b$</annotation> </semantics></math> blocks of size <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math> where every pair from <math> <semantics> <mrow> <mi>X</mi> </mrow> <annotation> $X$</annotation> </semantics></math> must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given <math> <semantics> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> </mrow> <annotation> $v,k$</annotation> </semantics></math>. In this paper, we use a new recursive construction to solve the existence of circular SCCD(<math> <semantics> <mrow> <mi>v</mi> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mi>b</mi> </mrow> <annotation> $v,4,b$</annotation> </semantics></math>) for all <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math> and three residue classes of circular SCCD(<math> <semantics> <mrow> <mi>v</mi> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mi>b</mi> </mrow> <annotation> $v,5,b$</annotation> </semantics

单个变更覆盖设计（SCCD）是一个v$v$集X$X$和一个有序列表ℒ $k$k$大小的b$b$块的{rm{mathcalL}}$，其中X$X$中的每一对必须出现在至少一个块中。每对连续块的不同之处仅在于一个元素。这是一种线性的单一变更覆盖设计，或者更简单地说，是一种单一变更覆盖的设计。当第一块和最后一块也因一个元素而不同时，单个变更覆盖设计是圆形的。如果不能为给定的v，k$v，k$构造其他较小的设计，则单个变更覆盖设计是最小的。在本文中，我们使用一个新的递归构造来求解所有v$v$的循环SCCD（v，4，b$v，4、b$）的存在性以及循环SCCD的三个残基类（v，5，b$v，5、b$）模16。我们求解了SCCD（v，5，b）的三个残基类的存在性$（v，5，b）$模16。我们证明了循环SCCD（2c）的存在性（k−1）+1，c 2（2 k−2）+c）$（2c（k-1）+1，k，｛c｝^｛2｝（2k-2）+c）$，对于所有c≥1，k≥2$cge 1，kge 2$，使用差分法。

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引用次数: 0

Self-dual association schemes, fusions of Hamming schemes, and partial geometric designs 自对偶关联方案、Hamming方案的融合和部分几何设计

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-05-25 DOI: 10.1002/jcd.21889

Bangteng Xu

Partial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self-dual association schemes. Then using the characterizations of self-dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self-dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2-designs. In particular, for a primitive self-dual symmetric association scheme <math> <semantics> <mrow> <mi>X</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>,</mo> <msub> <mrow> <mo>{</mo> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>}</mo> </mrow> <mrow> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> ${mathscr{X}}=(X,{{{R}_{i}}}_{0le ile 3})$</annotation> </semantics></math> of rank 4, if <math> <semantics> <mrow> <mo>∣</mo> <mi>X</mi> <mo>∣</mo> </mrow> <annotation> $| X| $</annotation> </semantics></math> is a power of 3 and each of <math> <semantics> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> </mrow> <annotation> $

局部几何设计可以由关联方案的基本关系来构造。Nowak等人（2016）根据某些Hamming方案的融合方案构建了一个无限族的局部几何设计。Xu（2023）给出了一种从关联方案创建局部几何设计的通用方法。在本文中，我们继续徐（2023）的研究。我们将首先研究自对偶关联方案的性质和特征。然后利用自对偶关联方案的特征和交换关联方案的表示理论（特征表），我们得到了秩为4的自对偶（对称或非对称）关联方案的刻画和分类，这些方案产生尽可能多的非平凡部分几何设计或2-设计。特别地，对于基元自对偶对称关联方案X＝（X，｛R i｝0≤i≤3）$｛mathscr｛X｝｝＝（X{{{R}_{i} ｛0le ile 3｝）$，如果｜X｜$|X|$是3的幂，并且R中的每一个为1${R}_{1} $，R 2${R}_{2} $和R0õR3${R}_｛0｝杯{R}_{3} $引入了部分几何设计，则我们将证明X$｛mathscr｛X｝｝$代数同构于Hamming方案H的一个融合方案（d，3）$H（d，三）$对于某个奇数d$d$。

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引用次数: 1

Primitive C 4 ${C}_{4}$ -decompositions of K n − I ${K}_{n}-I$ 基元C4${C}_{4} Kn−I的$-分解${K}_{n}-I$

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-05-05 DOI: 10.1002/jcd.21887

Michael W. Schroeder

A decomposition $� � � C � � �$ of a graph $� � � G � � �$ is primitive if no proper, nontrivial subset of $� � � C � � �$ is a decomposition of an induced subgraph of $� � � G � � �$ . An unresolved question posed by Asplund et al. in a recent publication involves the existence of primitive decompositions of cocktail party graphs into cycles of length 4, which is resolved by this paper.

图G$G$的分解C$｛mathscr｛C｝｝$是基元的，如果不是适当的，C$的非平凡子集是G$G$的诱导子图的分解。Asplund等人在最近的一篇出版物中提出了一个尚未解决的问题，涉及鸡尾酒会图到长度为4的循环的原始分解的存在性，本文对此进行了解决。

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引用次数: 0

Magic partially filled arrays on abelian groups 阿贝尔群上的魔术部分填充数组

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-05-04 DOI: 10.1002/jcd.21886

Fiorenza Morini, Marco Antonio Pellegrini

In this paper we introduce a special class of partially filled arrays. A magic partially filled array <math> <semantics> <mrow> <msub> <mtext>MPF</mtext> <mi>Ω</mi> </msub> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>s</mi> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> <annotation> ${text{MPF}}_{{rm{Omega }}}(m,n;s,k)$</annotation> </semantics></math> on a subset <math> <semantics> <mrow> <mi>Ω</mi> </mrow> <annotation> ${rm{Omega }}$</annotation> </semantics></math> of an abelian group <math> <semantics> <mrow> <mo>(</mo> <mi>Γ</mi> <mo>,</mo> <mo>+</mo> <mo>)</mo> </mrow> <annotation> $({rm{Gamma }},+)$</annotation> </semantics></math> is a partially filled array of size <math> <semantics> <mrow> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> <annotation> $mtimes n$</annotation> </semantics></math> with entries in <math> <semantics> <mrow> <mi>Ω</mi> </mrow> <annotation> ${rm{Omega }}$</annotation> </semantics></math> such that (i) every <math> <semantics> <mrow> <mi>ω</mi> <mo>∈</mo> <mi>Ω</mi> </mrow> <annotation> $omega in {rm{Omega }}$</annotation> </semantics></math> appears once in the array; (ii) each row contains <math> <semantics> <mrow> <mi>s</mi> </mrow> <annotation> $s$</annotation> </semantics></math> filled cells and each column contains <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math> filled cells; (iii) there exist (not necessarily distinct) elements <math> <semantics> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo>

本文介绍了一类特殊的部分填充数组。一种神奇的部分填充阵列MPFΩ子集Ω上的（m，n；s，k）$阿贝尔群（Γ，+）$（｛rm｛Gamma｝｝，+×n$mtimes n$中的条目为Ω$｛rm｛Omega｝｝$，使得（i）每个ω∈Ω$Omegain｛rm｛Omega｝｝$在数组中出现一次；（ii）每行包含s$s$填充单元格，每列包含k$k$填充单元格；（iii）存在（不一定不同）元素x、y∈Γ$x，yin｛rm｛Gamma｝｝$，使得每行中元素的和为x$x$，而每列中元素的总和为y$y$。特别地，如果x=y=0Γ$x=y={0}_｛rm｛Gamma｝｝$，我们有一个零和魔术部分填充阵列MPFΩ0（m，n；s，k）$｛｝^｛0｝text{MPF}_｛rm｛Omega｝｝（m，n；s，k）$。这些对象的例子有魔术矩形、Γ$｛rm｛Gamma｝｝$魔术矩形、有符号魔术数组、（整数或非整数）Heffter数组。在这里，我们给出了一个具有空单元格的魔术矩形存在的充要条件，即，MPFΩ（m，n；s，k）$｛text｛MPF｝｝_｛rm｛Omega｝｝｝（m，n；s，k）$其中Ω=｛1，2，…，n k｝⊂ℤ $｛rm｛Omega｝｝＝｛1,2，ldots，nk｝subet｛rm{mathbb｛Z｝｝｝｝$。当Ω$｛rm｛Omega｝｝$是阿贝尔群Γ$或其非零元素的集合时，我们还构造了零和魔术部分填充数组。

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引用次数: 0

On an Assmus–Mattson type theorem for type I and even formally self-dual codes 关于I型甚至形式自对偶码的Assmus–Mattson型定理

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-04-17 DOI: 10.1002/jcd.21883

Tsuyoshi Miezaki, Hiroyuki Nakasora

In the present paper, we give an Assmus–Mattson type theorem for near-extremal Type I and even formally self-dual codes. We show the existence of 1-designs or 2-designs for these codes. As a corollary, we prove the uniqueness of a self-orthogonal 2- $� � � (� � � 16 � �, � � 6 � �, � � 8 � � �) � � �$ design.

本文给出了近极值I型甚至形式自对偶码的一个Assmus–Mattson型定理。我们证明了这些代码的1-设计或2-设计的存在性。作为推论，我们证明了自正交2-（16，6，8）的唯一性$（16,6,8）$设计。

引用次数: 7

Ordered covering arrays and upper bounds on covering codes 有序覆盖数组与覆盖码的上界

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-03-30 DOI: 10.1002/jcd.21882

André Guerino Castoldi, Emerson L. Monte Carmelo, Lucia Moura, Daniel Panario, Brett Stevens

This work shows several direct and recursive constructions of ordered covering arrays (OCAs) using projection, fusion, column augmentation, derivation, concatenation, and Cartesian product. Upper bounds on covering codes in Niederreiter–Rosenbloom–Tsfasman (shorten by NRT) spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets. We give tables comparing the new upper bounds for covering codes to existing ones.

这项工作展示了使用投影、融合、列扩充、推导、级联和笛卡尔乘积的有序覆盖阵列（OCA）的几种直接和递归构造。通过改进一般上界，得到了Niederreiter–Rosenbloom–Tsfasman（用NRT缩短）空间中覆盖码的上界。我们探索了NRT空间中有序覆盖数组和覆盖码之间的联系，这推广了Hamming度量的类似结果。结合NRT空间中覆盖码的新上界和有序覆盖数组，我们改进了较大字母的NRT空间覆盖码的上界。我们给出了将覆盖码的新上界与现有上界进行比较的表格。

引用次数: 0

The existence of λ $lambda $ -decomposable super-simple ( 4 , 2 λ ) $(4,2lambda )$ -GDDs of type g u ${g}^{u}$ with λ = 2 , 4 $lambda =2,4$ λ$lambda$-可分解超简单（4,2λ）$（4,2lambda）$-GDD的存在性类型g u$｛g｝^｛u｝$，λ=2，4$lambda=2，4$

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-03-30 DOI: 10.1002/jcd.21881

Huangsheng Yu, Jingyuan Chen, R. Julian R. Abel, Dianhua Wu

A design is said to be super-simple if the intersection of any two of its blocks has at most two elements. A design with index <math> <semantics> <mrow> <mi>t</mi> <mi>λ</mi> </mrow> <annotation> $tlambda $</annotation> </semantics></math> is said to be <math> <semantics> <mrow> <mi>λ</mi> </mrow> <annotation> $lambda $</annotation> </semantics></math>-decomposable, if its blocks can be partitioned into nonempty collections <math> <semantics> <mrow> <msub> <mi>ℬ</mi> <mi>i</mi> </msub> </mrow> <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation> </semantics></math>, <math> <semantics> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>t</mi> </mrow> <annotation> $1le ile t$</annotation> </semantics></math>, such that each <math> <semantics> <mrow> <msub> <mi>ℬ</mi> <mi>i</mi> </msub> </mrow> <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation> </semantics></math> with the point set forms a design with index <math> <semantics> <mrow> <mi>λ</mi> </mrow> <annotation> $lambda $</annotation> </semantics></math>. In this paper, it is proved that for <math> <semantics> <mrow> <mi>λ</mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mn>2</mn> <mo>,</mo> <mn>4</mn> </mrow> <mo>}</mo> </mrow> </mrow> <annotation> $lambda in {2,4}$</annotation> </semantics></math>, there exists a <math> <semantics> <mrow> <mi>λ</mi> </mrow> <annotation> $lambda $</annotation> </semantics></math>-decomposable super-simple <math> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $(4,2lambda )$</annotation>

本文证明了对于λ∈{2，4}$lambdaIn{2,4}$，存在一个λ$lambda$-可分解超简单（4,2λ）$（4,2lambda）$-类型为g u$｛g｝^｛u｝$的GDD当且仅当u≥4$uge 4$，g（u−2）≥2λ$g（u-2）ge 2λ$和g（u−1）lect 0（mod 3）$g（u-1）equiv，0，（mathrm｛mod｝，3）$，除（g，u，λ）=（3，5，2）$（g，u，lambda）=（3,5,2）$，并且可能除了（g，u，λ）∈{（2，7，2），（6，5，4）}$（g，u，lambda）在｛（2,7,2），（6,5,4）｝$中。

{"title":"The existence of \u0000 \u0000 \u0000 λ\u0000 \u0000 $lambda $\u0000 -decomposable super-simple \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 4\u0000 ,\u0000 2\u0000 λ\u0000 \u0000 )\u0000 \u0000 \u0000 $(4,2lambda )$\u0000 -GDDs of type \u0000 \u0000 \u0000 \u0000 g\u0000 u\u0000 \u0000 \u0000 ${g}^{u}$\u0000 with \u0000 \u0000 \u0000 λ\u0000 =\u0000 2\u0000 ,\u0000 4\u0000 \u0000 $lambda =2,4$","authors":"Huangsheng Yu, Jingyuan Chen, R. Julian R. Abel, Dianhua Wu","doi":"10.1002/jcd.21881","DOIUrl":"https://doi.org/10.1002/jcd.21881","url":null,"abstract":"A design is said to be super-simple if the intersection of any two of its blocks has at most two elements. A design with index <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $tlambda $</annotation>\u0000 </semantics></math> is said to be <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math>-decomposable, if its blocks can be partitioned into nonempty collections <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ℬ</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>≤</mo>\u0000 <mi>i</mi>\u0000 <mo>≤</mo>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $1le ile t$</annotation>\u0000 </semantics></math>, such that each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ℬ</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation>\u0000 </semantics></math> with the point set forms a design with index <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math>. In this paper, it is proved that for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $lambda in {2,4}$</annotation>\u0000 </semantics></math>, there exists a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math>-decomposable super-simple <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(4,2lambda )$</annotation>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 6","pages":"289-303"},"PeriodicalIF":0.7,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50148436","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

Towards the Ryser–Woodall λ $lambda $ -design conjecture Ryser–Woodallλ$lambda$设计猜想

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-02-26 DOI: 10.1002/jcd.21878

Navin M. Singhi, Mohan S. Shrikhande, Rajendra M. Pawale

Let <math> <semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <annotation> ${r}_{1}$</annotation> </semantics></math> and <math> <semantics> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>></mo> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> ${r}_{2},({r}_{1}gt {r}_{2})$</annotation> </semantics></math> be the two replication numbers of a <math> <semantics> <mrow> <mi>λ</mi> </mrow> <annotation> $lambda $</annotation> </semantics></math>-design <math> <semantics> <mrow> <mi>D</mi> </mrow> <annotation> $D$</annotation> </semantics></math>. We denote the block size <math> <semantics> <mrow> <mo>∣</mo> <msub> <mi>B</mi> <mi>j</mi> </msub> <mo>∣</mo> </mrow> <annotation> $| {B}_{j}| $</annotation> </semantics></math> by <math> <semantics> <mrow> <msub> <mi>k</mi> <mi>j</mi> </msub> </mrow> <annotation> ${k}_{j}$</annotation> </semantics></math> and by <math> <semantics> <mrow> <msubsup> <mi>k</mi> <mi>j</mi> <mo>′</mo> </msubsup> </mrow> <annotation> ${k}_{j}^{^{prime} }$</annotation> </semantics></math> (respectively, <math> <semantics> <mrow> <msubsup> <mi>k</mi>

设r 1${r}_{1} $和r2，（r1&gt；r2）${r}_{2} ({r}_{1} gt{r}_{2} ）$是λ$lambda$设计D$D$的两个复制数。我们表示块大小ŞBjŞ$|{B}_{j} |$by kj${k}_{j} $和by k j′${k}_{j} ^｛^｛prime｝｝$（分别为kj*${k}_{j} ^｛*｝$）复制编号为r 1的点数${r}_{1} $（分别，r 2${r}_{2} $）${B}_{j} $$D$D$。取g=gcd r 1−r 2 gcd（r 1−1.r 2−1），λ，λ= 设r 1${r}_{1} $和r2，（r1&gt；r2）${r}_{2} ({r}_{1} gt{r}_{2} ）$是λ$lambda$设计D$D$的两个复制数。我们表示块大小ŞBjŞ$|{B}_{j} |$by kj${k}_{j} $和by k j′${k}_{j} ^｛^｛prime｝｝$（分别为kj*${k}_{j} ^｛*｝$）复制编号为r 1的点数${r}_{1} $（分别，r 2${r}_{2} $）${B}_{j} $$D$D$。

{"title":"Towards the Ryser–Woodall \u0000 \u0000 \u0000 λ\u0000 \u0000 $lambda $\u0000 -design conjecture","authors":"Navin M. Singhi, Mohan S. Shrikhande, Rajendra M. Pawale","doi":"10.1002/jcd.21878","DOIUrl":"https://doi.org/10.1002/jcd.21878","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>r</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${r}_{1}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>r</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>r</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <msub>\u0000 <mi>r</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${r}_{2},({r}_{1}gt {r}_{2})$</annotation>\u0000 </semantics></math> be the two replication numbers of a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math>-design <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math>. We denote the block size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <msub>\u0000 <mi>B</mi>\u0000 \u0000 <mi>j</mi>\u0000 </msub>\u0000 \u0000 <mo>∣</mo>\u0000 </mrow>\u0000 <annotation> $| {B}_{j}| $</annotation>\u0000 </semantics></math> by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>k</mi>\u0000 \u0000 <mi>j</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${k}_{j}$</annotation>\u0000 </semantics></math> and by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>k</mi>\u0000 \u0000 <mi>j</mi>\u0000 \u0000 <mo>′</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation> ${k}_{j}^{^{prime} }$</annotation>\u0000 </semantics></math> (respectively, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>k</mi>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 5","pages":"267-276"},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50144158","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

Alternating groups and point-primitive linear spaces with number of points being squarefree 交替群与点数为平方的点基元线性空间

IF 0.7 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2023-02-26 DOI: 10.1002/jcd.21879

Haiyan Guan, Shenglin Zhou

This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let $� � � S � � �$ be a nontrivial finite regular linear space whose number of points $� � � v � � �$ is squarefree. We prove that if $� � � G � \leq � Aut � � (� S �) � � � �$ is point-primitive with an alternating socle, then $� � � S � � �$ is the projective space $� � � PG � � (� � 3 �, � 2 � �) � � � �$ .

本文对点基元有限正则线性空间的分类作了进一步的贡献。设S$｛mathscr｛S｝｝$是一个非平凡的有限正则线性空间，其点数v$v$为平方。我们证明了如果G≤Aut（S）$Gletext｛Aut｝（｛mathscr｛S｝｝）$是具有交替socle的点基元，则S$｛mathscr｛S｝｝$是投影空间PG（3，2）$text｛PG｝（3,2）$。

{"title":"Alternating groups and point-primitive linear spaces with number of points being squarefree","authors":"Haiyan Guan, Shenglin Zhou","doi":"10.1002/jcd.21879","DOIUrl":"https://doi.org/10.1002/jcd.21879","url":null,"abstract":"This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math> be a nontrivial finite regular linear space whose number of points <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math> is squarefree. We prove that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>≤</mo>\u0000 <mtext>Aut</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $Gle text{Aut}({mathscr{S}})$</annotation>\u0000 </semantics></math> is point-primitive with an alternating socle, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math> is the projective space <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(3,2)$</annotation>\u0000 </semantics></math>.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 5","pages":"277-286"},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50144159","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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