Journal of Combinatorial Theory Series A最新文献

英文中文

Full weight spectrum one-orbit cyclic subspace codes

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-12-31 DOI: 10.1016/j.jcta.2024.106005

Chiara Castello, Olga Polverino, Ferdinando Zullo

For a linear Hamming metric code of length n over a finite field, the number of distinct weights of its codewords is at most n. The codes achieving the equality in the above bound were called full weight spectrum codes. In this paper, we will focus on the analogous class of codes within the framework of cyclic subspace codes. Cyclic subspace codes have garnered significant attention, particularly for their applications in random network coding to correct errors and erasures. We investigate one-orbit cyclic subspace codes that are full weight spectrum in this context. Utilizing number-theoretical results and combinatorial arguments, we provide a complete classification of full weight spectrum one-orbit cyclic subspace codes.

引用次数: 0

Contributions to Ma's conjecture concerning abelian difference sets with multiplier −1 (I)

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-12-31 DOI: 10.1016/j.jcta.2024.106004

Yasutsugu Fujita , Maohua Le

<div><div>Let <math><mi>N</mi></math>, <math><mi>P</mi></math> be the sets of all positive integers and odd primes, respectively. In 1991, when studying the existence of abelian difference sets with multiplier −1, S.-L. Ma [14] conjectured that the equation <math><mo>(</mo><mo>⁎</mo><mo>)</mo></math> <math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn></math>, <math><mi>p</mi><mo>∈</mo><mi>P</mi><mo>,</mo><mspace></mspace><mi>x</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math> has only one solution <math><mo>(</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>5</mn><mo>,</mo><mn>49</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>. This is a far from solved problem that has been poorly known for so long. In this paper, using some elementary methods, we first prove that if <math><mo>(</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math> is a solution of <math><mo>(</mo><mo>⁎</mo><mo>)</mo></math> with <math><mi>m</mi><mo>=</mo><mn>2</mn><mi>n</mi></math>, then there exist an odd positive integer g and a positive integer t which make <math><msup><mrow><mn>2</mn></mrow><mrow><mi>a</mi></mrow></msup><mo>=</mo><mo>(</mo><mi>g</mi><mo>+</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><mo>(</mo><mi>g</mi><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math> and <math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub></math>, where <math><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mo>(</mo><msup><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>/</mo><mo>(</mo><mi>α</mi><mo>−</mo><mover><mrow><mi>α</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math> for any integer r, <math><mi>α</mi><mo>=</mo><mn>2</mn><mi>g</mi><mo>+</mo><msqrt><mrow><mn>4</mn><msup><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt

引用次数: 0

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 0.9 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

F_{2}^{n}

, there should be Sidon sets of size at least

2^{n / 2} + 1

for all n. This paper provides an overview of the known large Sidon sets in

F_{2}^{n}

, 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。

引用次数: 0

Diametric problem for permutations with the Ulam metric (optimal anticodes)

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-12-19 DOI: 10.1016/j.jcta.2024.106002

Pat Devlin, Leo Douhovnikoff

We study the diametric problem (i.e., optimal anticodes) in the space of permutations under the Ulam distance. That is, let

S_{n}

denote the set of permutations on n symbols, and for each

σ, τ \in S_{n}

, define their Ulam distance as the number of distinct symbols that must be deleted from each until they are equal. We obtain a near-optimal upper bound on the size of the intersection of two balls in this space, and as a corollary, we prove that a set of diameter at most k has size at most

2^{k + C k^{2 / 3}} n! / (n - k)!

, compared to the best known construction of size

n! / (n - k)!

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引用次数: 0

Cayley extensions of maniplexes and polytopes

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-12-18 DOI: 10.1016/j.jcta.2024.106000

Gabe Cunningham , Elías Mochán , Antonio Montero

A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define

M

to be a Cayley extension of

K

if the facets of

M

are isomorphic to

K

and if some subgroup of the automorphism group of

M

acts regularly on the facets of

M

. We show that many natural extensions in the literature on maniplexes and polytopes are in fact Cayley extensions. We also describe several universal Cayley extensions. Finally, we examine the automorphism group and symmetry type graph of Cayley extensions.

引用次数: 0

On the size of integer programs with bounded non-vanishing subdeterminants

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2024-12-18 DOI: 10.1016/j.jcta.2024.106003

Björn Kriepke, Gohar M. Kyureghyan, Matthias Schymura

Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank r and with bounded subdeterminants. In particular, we study the column number question for integer matrices whose every

r \times r

minor is non-zero and bounded by a fixed constant Δ in absolute value. Approaching the problem in two different ways, one that uses results from coding theory, and the other from the geometry of numbers, we obtain linear and asymptotically sublinear upper bounds on the maximal number of columns of such matrices, respectively. We complement these results by lower bound constructions, matching the linear upper bound for

r = 2

, and a discussion of a computational approach to determine the maximal number of columns for small parameters Δ and r.

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 0.9 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

η^{N} (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

η^{N} (C_{n} \oplus C_{n}) = 3 n + 1

for every

n \geq 2

and solved the corresponding inverse problem for groups

C_{p} \oplus C_{p}

, where p is a prime. In this paper, we determine the precise value of

η^{N} (G)

for all finite abelian groups of rank 2 and resolve the corresponding inverse problem for groups

C_{n} \oplus C_{n}

, where

n \geq 2

, which confirms a conjecture of Gao, Geroldinger and Wang for all

n \geq 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。

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