Journal of Combinatorial Designs最新文献_第4页

On the Terwilliger Algebra of the Group Association Scheme of the Symmetric Group Sym ( 7 ) 关于对称群Sym群关联方案的Terwilliger代数(7)

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-04-03 DOI: 10.1002/jcd.21981

Allen Herman, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra

Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups $� � � � � Sym � � � (� � n � �) � � � �$ , for $� � � � � 3 � � \leq � � n � � \leq � � 6 � � �$ , have been studied and completely determined. The case for $� � � � � Sym � � � (� � 7 � �) � � � �$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $� � � � � Sym � � � (� � 7 � �) � � � �$ with a minimal distance of at least 4. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group $� � � � � Sym � � � (� �$

Terwilliger代数是有限维半简单代数，由Paul Terwilliger于1992年在关联方案和距离正则图的研究中首次引入。对称群Sym (n)的共轭类关联方案的Terwilliger代数对于3≤n≤6，已经研究并完全确定。Sym(7)的情况在计算上要困难得多，并且有一个潜在的应用程序来查找最大排列代码的大小Sym(7)的最小距离至少为4。在本文中，维数，Wedderburn分解，确定了群Sym(7)的共轭类格式的Terwilliger代数的块维分解。

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引用次数: 0

The Directed Oberwolfach Problem With Variable Cycle Lengths: A Recursive Construction 变周期长的有向Oberwolfach问题：一个递归构造

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-03-24 DOI: 10.1002/jcd.21967

Suzan Kadri, Mateja Šajna

The directed Oberwolfach problem <math> <semantics> <mrow> <mrow> <msup> <mstyle> <mspace></mspace> <mtext>OP</mtext> <mspace></mspace> </mstyle> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> asks whether the complete symmetric digraph <math> <semantics> <mrow> <mrow> <msubsup> <mi>K</mi> <mi>n</mi> <mo>*</mo> </msubsup> </mrow> </mrow> </semantics></math>, assuming <math> <semantics> <mrow> <mrow> <mi>n</mi> <mo>=</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>⋯</mi> <mo>+</mo> <msub> <mi>m</mi> <mi>k</mi> </msub> </mrow> </mrow> </semantics></math>, admits a decomposition into spanning subdigraphs, each a disjoint un

有向Oberwolfach问题OP * (m1，…k)问是否完全对称有向图K n *，假设n = m1 +⋯+k，允许分解成生成的子图，每个都是k个长度为m1的有向循环的不相交并，…，我…在此，我们描述了构造OP *（）的解的方法。m1，…M (k)给出的解OP * (m 1 有向Oberwolfach问题OP * (m1，…k)问是否完全对称有向图K n *，假设n = m1 +⋯+k，允许分解成生成的子图，每个都是k个长度为m1的有向循环的不相交并，…，我… 在此，我们描述了构造OP *（）的解的方法。m1，…M (k)给出的解OP * (m 1，……M (l)，对于一些，l &lt；K，如果在m1上的某些条件，M k是满足的。

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引用次数: 0

Putatively Optimal Projective Spherical Designs With Little Apparent Symmetry 具有少量明显对称性的推定最优射影球面设计

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-03-11 DOI: 10.1002/jcd.21979

Alex Elzenaar, Shayne Waldron

We give some new explicit examples of putatively optimal projective spherical designs, that is, ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in general, which requires the introduction of new techniques for their construction. New examples of interest include an 11-point spherical

� � �                     � � � (�                           � � 3 �                              �, �                              � 3 � �                           �) � � � �

-design for

� � �                     � � �^{R �                           � 3 �} � � �

, and a 12-point spherical

� � �                     � � � (�                           � � 2 �                              �, �                              � 2 � �                           �) � � � �

-design for

� � �                     � � �^{R �                           � 4 �} � � �

given by four Mercedes-Benz frames that lie on equi-isoclinic planes. The latter example shows that the set of optimal spherical designs can be uncountable. We also give results of an extensive numerical study to determine the nature of the real algebraic variety of optimal projective real spherical designs, and in particular when it is a single point (a unique design) or corresponds to an infinite family of designs.

我们给出了一些新的明确的假设最优射影球面设计的例子，即那些有数值证据证明它们是最小尺寸的。这些形成连续的家族，因此通常没有明显的对称性，这就需要引入新的建造技术。新的有趣的例子包括11点球面(3)，3) R 3的设计；一个12点球面(2)2)给出了r4的设计由四个梅赛德斯-奔驰框架组成，它们位于等斜平面上。后一个例子表明最优球面设计的集合可以是不可数的。我们还给出了广泛的数值研究结果，以确定最优射影实球面设计的实代数变化的性质，特别是当它是单点（唯一设计）或对应于无限族的设计时。

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引用次数: 0

A Classification of the Flag-Transitive 2- ( v , 3 , λ ) Designs With v ≡ 1 , 3 ( mod 6 ) and v ≡ 6 ( mod λ ) 一类具有v≡1的2- (v, 3, λ)标志传递设计3 （mod 6）和v≡6 (modλ )

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-03-05 DOI: 10.1002/jcd.21978

Alessandro Montinaro, Eliana Francot

<div> In this paper, we provide a complete classification of the 2-<math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> designs with <math> <semantics> <mrow> <mrow> <mi>v</mi> <mo>≡</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mspace></mspace> <mrow> <mo>(</mo> <mrow> <mi>mod</mi> <mspace></mspace> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> and <math> <semantics> <mrow> <mrow> <mi>v</mi> <mo>≡</mo> <mn>6</mn> <mspace></mspace> <mrow> <mo>(</mo> <mrow> <mi>mod</mi> <mspace></mspace> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> admitting a flag-transitive automorphism group non-isomorphic to a subgroup of <math> <semantics> <mrow> <mrow> <mi>A</mi> <mi>Γ</mi> <msub> <mi>L</mi> <mn>1</mn> </msub>

在本文中，我们提供了2- (v, 3，λ)设计v≡1，3 （mod 6）和V≡6 （mod λ）承认一个与a的子群Γ L 1 (v)不同构的标志传递自同构群) .

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引用次数: 0

GDD Type Spanning Bipartite Block Designs GDD类型跨越二部块设计

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-02-27 DOI: 10.1002/jcd.21976

Shoko Chisaki, Ryoh Fuji-Hara, Nobuko Miyamoto

<div> There is a one-to-one correspondence between the point set of a group divisible design (GDD) with <math> <semantics> <mrow> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> </mrow> </mrow> </semantics></math> groups of <math> <semantics> <mrow> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </mrow> </semantics></math> points and the edge set of a complete bipartite graph <math> <semantics> <mrow> <mrow> <msub> <mi>K</mi> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </mrow> </semantics></math>. A block of GDD corresponds to a subgraph of <math> <semantics> <mrow> <mrow> <msub> <mi>K</mi> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </mrow> </semantics></math>. We show that the concurrence conditions of two points of GDD can correspond to the edge concurrence conditions of subgraphs of <math> <semantics> <mrow> <mrow> <msub> <mi>K</mi> <mrow> <msub>

群可分设计（GDD）的点集与v 2的v 1组之间存在一一对应关系点和完全二部图kv1的边集，V 2。GDD的一个块对应于kv1的一个子图，V 2。我们证明了GDD两点的并发条件可以对应于kv1的子图的边并发条件，我们称之为gdd型跨二部块设计（SBBD）。我们还提出了一种直接从(r)构造SBBD的方法，λ) -设计和群上的差分矩阵。当an (r)λ) -设计与v点有b块大得多而v，提出了一种改进的方法来构造一个块数更少的SBBD，使v1更接近v2通过划分(r)的块集，λ) -设计。

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引用次数: 0

Cycle Switching in Steiner Triple Systems of Order 19 19阶Steiner三重系统的循环切换

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-02-27 DOI: 10.1002/jcd.21975

Grahame Erskine, Terry S. Griggs

<div> Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order <math> <semantics> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> </semantics></math> (an <math> <semantics> <mrow> <mrow> <mtext>STS</mtext> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math>), yielding another <math> <semantics> <mrow> <mrow> <mtext>STS</mtext> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math>. This relationship may be represented by an undirected graph. An <math> <semantics> <mrow> <mrow> <mtext>STS</mtext> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> admits cycles of lengths <math> <semantics> <mrow> <mrow> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mi>…</mi> <mo>,</mo> <mi>v</mi> <mo>−</mo> <mn>7</mn> </mrow> </mrow> </semantics></math> and <math> <semantics> <mrow> <mrow> <mi>v</mi> <mo>−</mo> <mn>3</mn> </mrow> </mrow> </semantics></math>. In the particular case of <math> <semantics> <mrow>

循环交换是应用于给定阶v(和STS (v)的Steiner三重系统同构类的一种特殊变换形式))，产生另一个化粪池系统(v)。这种关系可以用无向图来表示。STS (v)允许长度为4的循环，6、……V−7和V−3。在v = 19的特殊情况下，已知允许任意长度的循环切换的全切换图是连通的。我们证明，如果我们只限制切换到一个可能的周期长度，在所有情况下，切换图是断开的（即使我们忽略那些STS (19) s），它们没有给定长度的循环)。此外，在许多情况下，我们发现在切换图中有趣的连接组件，它们表现出意想不到的对称性。我们的方法利用一种算法来确定一个非常大的隐式定义图中的连接组件，这比以前的方法更有效，避免了对大部分系统计算规范标记的必要性。

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引用次数: 0

More Heffter Spaces via Finite Fields 通过有限域得到更多的空间

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-02-23 DOI: 10.1002/jcd.21974

Marco Buratti, Anita Pasotti

<div> A <math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>;</mo> <mi>r</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> Heffter space is a resolvable <math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> configuration whose points form a half-set of an abelian group <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> and whose blocks are all zero-sum in <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math>. It was recently proved that there are infinitely many orders <math> <semantics> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> </semantics></math> for which, given any pair <math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mi>r</mi> </mrow>

A (v, k；hffter空间是一个可分辨的（v r）,b (k)位形，其点构成阿贝尔群G的半集它的方块在G中都是零和的。最近证明了存在无穷多个v阶，对于任意一对(k)R) k≥3奇数，A (v, k；r) Heffter空间存在。这是通过施加一个点正则自同构群得到的。在这里，我们通过要求一个点半正则自同构群来放宽这个要求。这样，上述结果也可以推广到k为偶数的情况。

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引用次数: 0

Pairs in Nested Steiner Quadruple Systems 嵌套Steiner四重系统中的对

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-02-20 DOI: 10.1002/jcd.21973

Yeow Meng Chee, Son Hoang Dau, Tuvi Etzion, Han Mao Kiah, Wenqin Zhang

Motivated by a repair problem for fractional repetition codes in distributed storage, each block of any Steiner quadruple system (SQS) of order $� � � � � v � � �$ is partitioned into two pairs. Each pair in such a partition is called a nested design pair, and its multiplicity is the number of times it is a pair in this partition. Such a partition of each block is considered a new block design called a nested SQS. Several related questions on this type of design are considered in this paper: What is the maximum multiplicity of the nested design pair with minimum multiplicity? What is the minimum multiplicity of the nested design pair with maximum multiplicity? Are there nested quadruple systems in which all the nested design pairs have the same multiplicity? Of special interest are nested quadruple systems in which all the $� � � � � (�, \frac{�}{v}, �) � � �$ pairs are nested design pairs with the same multiplicity. Several constructions of nested quadruple systems are considered, and in particular, classic constructions of SQS are examined.

针对分布式存储中分数阶重复码的修复问题，将任意v阶Steiner四重系统（SQS）的每个块划分为两对。这种分区中的每对设计对称为嵌套设计对，其多重性是在该分区中成为一对的次数。每个块的这样一个分区被认为是一个新的块设计，称为嵌套SQS。本文考虑了这类设计的几个相关问题：具有最小多重性的嵌套设计对的最大多重性是什么？具有最大多重性的嵌套设计对的最小多重性是什么？是否存在嵌套的四重系统，其中所有嵌套的设计对具有相同的多重性？特别有趣的是嵌套四重系统，其中所有的v2对都是具有相同多重性的嵌套设计对。考虑了嵌套四重系统的几种结构，特别是对SQS的经典结构进行了研究。

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引用次数: 0

New Families of Strength-3 Covering Arrays Using Linear Feedback Shift Register Sequences 基于线性反馈移位寄存器序列的新型强度-3覆盖阵列

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-02-14 DOI: 10.1002/jcd.21963

Kianoosh Shokri, Lucia Moura

In an array over an alphabet of <math> <semantics> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> </semantics></math> symbols, a <math> <semantics> <mrow> <mrow> <mi>t</mi> </mrow> </mrow> </semantics></math>-set of column indices <math> <semantics> <mrow> <mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>c</mi> <mi>t</mi> </msub> </mrow> <mo>}</mo> </mrow> </mrow> </mrow> </semantics></math> is covered if each <math> <semantics> <mrow> <mrow> <mi>t</mi> </mrow> </mrow> </semantics></math>-tuple of the alphabet occurs at least once as a row of the sub-array indexed by <math> <semantics> <mrow> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>c</mi> <mi>t</mi> </msub> </mrow> </mrow> </semantics></math>. A covering array, denoted by CA<math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>N</mi>

在一个包含v个符号的字母表数组中，列索引{c1}的t集合，……C t}被覆盖，如果每个t-tuple的字母表至少出现一次，作为以c1为索引的子数组的一行，…，选C。一个覆盖阵列，表示为CA (N)；T k v)，是一个N × k的数组在一个有v个符号的字母表上，它的性质是anyT -列的集合被覆盖。这里，N是覆盖阵列的大小，t是覆盖阵列的强度。Raaphorst等人（Des. Codes Cryptogr.）。（2014） 73:949-968)给出了CA （2 q 3−）的结构1 ;3 ,q2 + Q + 1，q)我们记作rq，利用特征多项式为fq上的原始多项式的线性反馈移位寄存器（LFSR）序列。数组rq对应于一个覆盖的完美哈希族。我们给出了一个基于水平连接rq的x个副本的覆盖强度为3的数组的构造，对于任意素数幂q和x∈{2， q, q + 1，问2：q2−Q + 1} .覆盖是通过开发roux型结构来完成的，这种结构利用了rq的结构并删除了重复的行。这些覆盖数组中的一些改进了先前已知的具有相同相应参数的覆盖数组的大小N的上界。

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引用次数: 0

On MSRD Codes, h-Designs and Disjoint Maximum Scattered Linear Sets 关于MSRD规范、h-设计和不相交最大离散线性集

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs

Pub Date : 2025-01-29 DOI: 10.1002/jcd.21972

Paolo Santonastaso, John Sheekey

In this paper, we construct new optimal subspace designs and, consequently, new optimal codes in the sum-rank metric. We construct new 1-designs by finding sets of disjoint maximum scattered linear sets, and use these constructions to also find new $� � � � h � � �$ -designs for $� � � � h � > � 1 � � �$ . As a means of achieving this, we establish a correspondence between the metric properties of sum-rank metric codes and the geometric properties of subspace designs. Specifically, we determine the geometric counterpart of the coding-theoretic notion of generalised weights for the sum-rank metric in terms of subspace designs and determine a geometric characterisation of MSRD codes. This enables us to characterise subspace designs via their intersections with hyperplanes and via duality operations.

在本文中，我们构造了新的最优子空间设计，从而在和秩度量中构造了新的最优码。我们通过寻找不相交的最大分散线性集来构造新的1-设计，并利用这些构造来寻找h >的新的h -设计；1 .为了达到这个目的，我们建立了和秩度量码的度量性质与子空间设计的几何性质之间的对应关系。具体来说，我们在子空间设计方面确定了和秩度量的广义权重的编码理论概念的几何对应物，并确定了MSRD码的几何特征。这使我们能够通过它们与超平面的交点和对偶运算来表征子空间设计。

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引用次数: 0