Journal of Combinatorial Theory Series A最新文献

英文中文

Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions 西顿集，瘦集，和向量布尔函数的非线性

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-12-19 DOI: 10.1016/j.jcta.2024.106001

Gábor P. Nagy

The vectorial nonlinearity of a vector-valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager, and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet established a lower bound in terms of differential uniformity. In this paper, we improve Carlet's lower bound. Our approach is based on the fact that the level sets of a vectorial Boolean function are thin sets. In particular, level sets of APN functions are Sidon sets, hence the Liu-Mesnager-Chen conjecture predicts that in F2n, there should be Sidon sets of size at least 2n/2+1 for all n. This paper provides an overview of the known large Sidon sets in F2n, and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane.

向量值函数的向量非线性是它到仿射函数集合的距离。2017年，Liu、Mesnager和Chen推测了向量线性的一般上界。最近，Carlet建立了微分均匀性的下界。本文改进了Carlet下界。我们的方法是基于这样一个事实，即向量布尔函数的水平集是瘦集。特别是，APN函数的水平集是Sidon集，因此，Liu-Mesnager-Chen猜想预测，在F2n中，对于所有n，应该存在大小至少为2n/2+1的Sidon集。本文概述了F2n中已知的大Sidon集，并检验了由有限仿射平面的双曲线和椭圆导出的大Sidon集的完备性。

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引用次数: 0

On joint short minimal zero-sum subsequences over finite abelian groups of rank two 二阶有限阿贝尔群上的联合短最小零和子序列

IF 1.1 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-12-03 DOI: 10.1016/j.jcta.2024.105984

Yushuang Fan, Qinghai Zhong

Let (G,+,0) be a finite abelian group and let ηN(G) be the smallest integer ℓ such that every sequence over G∖{0} of length ℓ has two joint short minimal zero-sum subsequences. In 2013, Gao et al. obtained that ηN(Cn⊕Cn)=3n+1 for every n≥2 and solved the corresponding inverse problem for groups Cp⊕Cp, where p is a prime. In this paper, we determine the precise value of ηN(G) for all finite abelian groups of rank 2 and resolve the corresponding inverse problem for groups Cn⊕Cn, where n≥2, which confirms a conjecture of Gao, Geroldinger and Wang for all n≥2 except n=4.

设（G,+,0）是一个有限的阿贝尔群，设ηN(G)是最小的整数，使得G +{0}上的每一个长度为r的序列都有两个联合的最小零和子序列。2013年，Gao等人得到了n≥2时ηN(Cn⊕Cn)=3n+1，并求解了相应的群Cp⊕Cp的逆问题，其中p为素数。本文确定了所有2阶有限阿贝耳群的ηN(G)的精确值，并解决了n≥2的群Cn⊕Cn的逆问题，证实了Gao、Geroldinger和Wang对除n=4外所有n≥2的猜想。

引用次数: 0

The degree of functions in the Johnson and q-Johnson schemes Johnson和q-Johnson格式中函数的度

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-29 DOI: 10.1016/j.jcta.2024.105979

Michael Kiermaier , Jonathan Mannaert , Alfred Wassermann

In 1982, Cameron and Liebler investigated certain special sets of lines in

PG (3, q)

, and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these Cameron-Liebler line classes got much attention. Several generalizations and variants have been considered in the literature, the main directions being a variation of the dimensions of the involved spaces, and studying the analogous situation in the subset lattice. An important tool is the interpretation of the objects as Boolean functions in the Johnson and q-Johnson schemes.

In this article, we develop a unified theory covering all these variations. Generalized versions of algebraic and geometric properties will be investigated, having a parallel in the notion of designs and antidesigns in association schemes. This leads to a natural definition of the degree and the weights of functions in the ambient scheme, refining the existing definitions. We will study the effect of dualization and of elementary modifications of the ambient space on the degree and the weights. Moreover, a divisibility property of the sizes of Boolean functions of degree t will be proven.

1982年，Cameron和Liebler研究了PG（3,q）中某些特殊的线集，并给出了几个等价的刻画。由于其有趣的几何和代数性质，这些卡梅伦-李伯勒线类得到了广泛的关注。在文献中已经考虑了几种推广和变体，主要方向是相关空间的维数的变化，并研究了子集格中的类似情况。一个重要的工具是将对象解释为Johnson和q-Johnson方案中的布尔函数。在本文中，我们将建立一个涵盖所有这些变化的统一理论。将研究代数和几何性质的广义版本，在关联方案的设计和反设计的概念中具有并行性。这导致了环境方案中函数的程度和权重的自然定义，改进了现有的定义。我们将研究对偶化和环境空间的初等修改对度和权的影响。此外，还证明了t次布尔函数大小的可整除性。

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引用次数: 0

Sequence reconstruction problem for deletion channels: A complete asymptotic solution 删除信道的序列重建问题：一个完整的渐近解

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-27 DOI: 10.1016/j.jcta.2024.105980

Van Long Phuoc Pham , Keshav Goyal , Han Mao Kiah

Transmit a codeword

, that belongs to an

(ℓ - 1)

-deletion-correcting code of length n, over a t-deletion channel for some

1 \leq ℓ \leq t < n

. Levenshtein (2001) [10], proposed the problem of determining

N (n, ℓ, t) + 1

, the minimum number of distinct channel outputs required to uniquely reconstruct

. Prior to this work,

N (n, ℓ, t)

is known only when

ℓ \in {1, 2}

. Here, we provide an asymptotically exact solution for all values of ℓ and t. Specifically, we show that

N (n, ℓ, t) = \frac{(\begin{matrix} 2 ℓ \\ ℓ \end{matrix})}{(t - ℓ)!} n^{t - ℓ} - O (n^{t - ℓ - 1})

. In the special instances: where

ℓ = t

, we show that

N (n, ℓ, ℓ) = (\begin{matrix} 2 ℓ \\ ℓ \end{matrix})

; and when

ℓ = 3

and

t = 4

, we show that

N (n, 3, 4) \leq 20 n - 150

. We also provide a conjecture on the exact value of

N (n, ℓ, t)

for all values of n, ℓ, and t.

在某个 1≤ℓ≤t<n 的 t 缺失信道上传输属于长度为 n 的 (ℓ-1)- 缺失校正码的编码词 , 。Levenshtein （2001 年）[10] 提出了确定 N(n,ℓ,t)+1（唯一重构所需的最小不同信道输出数）的问题。在这项工作之前，N(n,ℓ,t) 只有在 ℓ∈{1,2} 时才是已知的。具体来说，我们证明了 N(n,ℓ,t)=(2ℓℓ)(t-ℓ)!nt-ℓ-O(nt-ℓ-1)。在特殊情况下：当 ℓ=t 时，我们证明了 N(n,ℓ,ℓ)=(2ℓℓ)；当 ℓ=3 和 t=4 时，我们证明了 N(n,3,4)≤20n-150 。我们还对 N(n,ℓ,t)在所有 n、ℓ 和 t 值下的精确值提出了猜想。

引用次数: 0

Non-empty pairwise cross-intersecting families 非空成对交叉族

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-26 DOI: 10.1016/j.jcta.2024.105981

Yang Huang, Yuejian Peng

<div><div>Two families <math><mi>A</mi></math> and <math><mi>B</mi></math> are cross-intersecting if <math><mi>A</mi><mo>∩</mo><mi>B</mi><mo>≠</mo><mo>∅</mo></math> for any <math><mi>A</mi><mo>∈</mo><mi>A</mi></math> and <math><mi>B</mi><mo>∈</mo><mi>B</mi></math>. We call t families <math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub></math> pairwise cross-intersecting families if <math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and <math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math> are cross-intersecting for <math><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>t</mi></math>. Additionally, if <math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math> for each <math><mi>j</mi><mo>∈</mo><mo>[</mo><mi>t</mi><mo>]</mo></math>, then we say that <math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub></math> are non-empty pairwise cross-intersecting. Let <math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow></math> be non-empty pairwise cross-intersecting families with <math><mi>t</mi><mo>≥</mo><mn>2</mn></math>, <math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>t</mi></mrow></msub></math>, <math><mi>n</mi><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and <math><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></m

如果对于任意的 A∈A 和 B∈B 来说，A∩B≠∅Sm_2205↩，则两个族 A 和 B 是相交的。如果 Ai 和 Aj 在 1≤i<j≤t 时交叉，我们称 t 个族为 A1,A2,...,At 成对交叉族。此外，如果对于每个 j∈[t] Aj≠∅，那么我们说 A1,A2,...At 是非空的成对相交族。设 A1⊆([n]k1),A2⊆([n]k2),...,At⊆([n]kt)为非空成对相交族，t≥2，k1≥k2≥⋯≥kt，n≥k1+k2，d1,d2,...,dt 为正数。本文给出了∑j=1tdj|Aj|的尖锐上界，并描述了达到上界的族 A1,A2,...At 的特征。我们的结果统一了 Frankl 和 Tokushige (1992) [5]、Shi、Frankl 和 Qian (2022) [15]、Huang、Peng 和 Wang [10] 以及 Zhang 和 Feng [16] 的结果。此外，我们的结果可以应用于对某些 n<k1+k2 的处理，而之前已知的所有结果都没有这样的应用。在证明过程中，我们应用了 Kruskal 和 Katona 的一个结果，使我们只考虑其元素是按词典顺序排列的第一个 |Ai| 元素的 Ai 族。我们用一个单变量函数 fi(R) 限定∑i=1tdi|Ai|，其中 R 是按词法顺序排列的 Ai 的最后一个元素，并验证了 -fi(R)具有比极值结果更强的单调性。我们认为，除了极值结果之外，本文中函数的单模态性本身也很有趣。

引用次数: 0

A classification of the flag-transitive 2-(v,k,2) designs 2-(v,k,2)旗转设计的分类

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-26 DOI: 10.1016/j.jcta.2024.105983

Hongxue Liang , Alessandro Montinaro

In this paper, we provide a complete classification of 2-

(v, k, 2)

designs admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear 1-dimensional group. Alongside this analysis, we provide a construction of seven new families of such flag-transitive 2-designs, one of them infinite, and some of them involving remarkable objects such as t-spreads, translation planes, quadrics and Segre varieties.

Our result together with those of Alavi et al. [1], [2], Praeger et al. [17], Zhou and the first author [39], [40] provides a complete classification of 2-

(v, k, 2)

design admitting a flag-transitive automorphism group with the only exception of the semilinear 1-dimensional case.

在本文中，我们提供了一个完整的 2-(v,k,2)设计分类，这些设计允许一个仿射类型的旗透式自变群，唯一的例外是半线性一维群。在进行分析的同时，我们还构建了七个新的旗透式 2-设计族，其中一个是无限设计族，其中一些设计族涉及诸如 t 展开、平移平面、四边形和 Segre varieties 等非凡对象。我们的结果与 Alavi 等人[1], [2], Praeger 等人[17], Zhou 和第一作者[39], [40]的结果一起，提供了一个完整的 2-(v,k,2) 设计的分类，其中只有半线性一维情况例外。

引用次数: 0

Distributions of reciprocal sums of parts in integer partitions 整数分区中各部分倒数之和的分布

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-26 DOI: 10.1016/j.jcta.2024.105982

Byungchan Kim , Eunmi Kim

Let

D_{n}

be the set of partitions of n into distinct parts, and

srp (λ)

be the sum of reciprocals of the parts of the partition λ. We show that as

n \to \infty

E [srp (λ) : λ \in D_{n}] \sim \frac{\log (3 n)}{4} and Var [srp (λ) : λ \in D_{n}] \sim \frac{π^{2}}{24} .

Moreover, for

P_{n}

, the set of ordinary partitions of n, we show that as

n \to \infty

E [srp (λ) : λ \in P_{n}] \sim π \sqrt{\frac{n}{6}} and Var [srp (λ) : λ \in P_{n}] \sim \frac{π^{2}}{15} n .

To prove these asymptotic formulas in a uniform manner, we derive a general asymptotic formula using Wright's circle method.

设 Dn 是将 n 分割为不同部分的集合，srp(λ) 是分割 λ 的各部分的倒数之和。我们证明，当 n→∞ 时，E[srp(λ):λ∈Dn]∼log(3n)4andVar[srp(λ):λ∈Dn]∼π224。此外，对于 n 的普通分区集合 Pn，我们证明当 n→∞ 时，E[srp(λ):λ∈Pn]∼πn6andVar[srp(λ):λ∈Pn]∼π215n。为了统一证明这些渐近公式，我们利用赖特圆法推导出一个一般渐近公式。

引用次数: 0

Dominance complexes, neighborhood complexes and combinatorial Alexander duals 支配复合体、邻域复合体和组合亚历山大对偶

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-16 DOI: 10.1016/j.jcta.2024.105978

Takahiro Matsushita , Shun Wakatsuki

We show that the dominance complex

D (G)

of a graph G coincides with the combinatorial Alexander dual of the neighborhood complex

N (\overline{G})

of the complement of G. Using this, we obtain a relation between the chromatic number

χ (G)

of G and the homology group of

D (G)

. We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest a new method for computing the homology groups of the dominance complexes, using independence complexes of simple graphs. We show that several known computations of homology groups of dominance complexes can be reduced to known computations of independence complexes. Finally, we determine the homology group of

D (P_{n} \times P_{3})

by determining the homotopy types of the independence complex of

P_{n} \times P_{3} \times P_{2}

我们证明了图 G 的支配复数 D(G) 与 G 的补集的邻域复数 N(G‾) 的组合亚历山大对偶重合，并由此得到了 G 的色度数 χ(G) 与 D(G) 的同调群之间的关系。我们还从邻接复数的著名事实中得到了几个与支配复数有关的已知结果。之后，我们提出了一种利用简单图的独立复数计算支配复数同调群的新方法。我们证明了支配复数同调群的几种已知计算方法可以简化为独立复数的已知计算方法。最后，我们通过确定 Pn×P3×P2 独立复数的同调类型来确定 D(Pn×P3) 的同调群。

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引用次数: 0

Upper bounds for the number of substructures in finite geometries from the container method 从容器法看有限几何中子结构数量的上界

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-06 DOI: 10.1016/j.jcta.2024.105968

Sam Mattheus, Geertrui Van de Voorde

We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial ovoids and EKR-sets of flags in polar spaces, line spreads in

PG (2 r - 1, q)

and plane spreads in

PG (5, q)

, and caps in

PG (3, q)

. The latter result extends work due to Roche-Newton and Warren [21] and Bhowmick and Roche-Newton [6].

Finally, we investigate caps in p-random subsets of

PG (r, q)

, which parallels recent work for arcs in projective planes by Bhowmick and Roche-Newton, and Roche-Newton and Warren [6], [21], and arcs in projective spaces by Chen, Liu, Nie and Zeng [8].

我们利用代数和极值组合学的技术，推导出有限几何中若干（超）图中独立集数的上界。通过这种方法，我们得到了极空间中部分敖包和旌旗的 EKR 集、PG(2r-1,q) 中的线展和 PG(5,q) 中的面展以及 PG(3,q) 中的盖的渐近尖锐上界。最后，我们研究了 PG(r,q) 的 p 个随机子集中的盖，这与 Bhowmick 和 Roche-Newton 以及 Roche-Newton 和 Warren [6], [21] 最近针对投影平面中的弧所做的工作，以及 Chen, Liu, Nie 和 Zeng [8] 最近针对投影空间中的弧所做的工作相似。

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引用次数: 0

The vector space generated by permutations of a trade or a design 由交易或设计的排列组合产生的向量空间

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-11-06 DOI: 10.1016/j.jcta.2024.105969

E. Ghorbani , S. Kamali , G.B. Khosrovshahi

Motivated by a classical result of Graver and Jurkat (1973) and Graham, Li, and Li (1980) in combinatorial design theory, which states that the permutations of t-

(v, k)

minimal trades generate the vector space of all t-

(v, k)

trades, we investigate the vector space spanned by permutations of an arbitrary trade. We prove that this vector space possesses a decomposition as a direct sum of subspaces formed in the same way by a specific family of so-called total trades. As an application, we demonstrate that for any t-

(v, k, λ)

design, its permutations can span the vector space generated by all t-

(v, k, λ)

designs for sufficiently large values of v. In other words, any t-

(v, k, λ)

design, or even any t-trade, can be expressed as a linear combination of permutations of a fixed t-design. This substantially extends a result by Ghodrati (2019), who proved the same result for Steiner designs.

受 Graver 和 Jurkat（1973 年）以及 Graham、Li 和 Li（1980 年）在组合设计理论中的一个经典结果（即 t-(v,k)最小交易的排列组合产生所有 t-(v,k)交易的向量空间）的启发，我们研究了任意交易的排列组合所跨越的向量空间。我们证明，这个向量空间可以分解为由特定的所谓总交易系列以相同方式形成的子空间的直接和。换句话说，任何 t-（v,k,λ）设计，甚至任何 t 交易，都可以表示为固定 t 设计的排列组合的线性组合。这大大扩展了 Ghodrati（2019）的一个结果，他为斯坦纳设计证明了同样的结果。

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引用次数: 0

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