Journal of Combinatorial Theory Series A最新文献

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

Journal of Combinatorial Theory Series A Pub Date : 2024-12-19 DOI: 10.1016/j.jcta.2024.106001

Gábor P. Nagy

{"title":"Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions","authors":"Gábor P. Nagy","doi":"10.1016/j.jcta.2024.106001","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.106001","url":null,"abstract":"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 <mml:math altimg=\"si1.svg\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"double-struck\">F</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math>, there should be Sidon sets of size at least <mml:math altimg=\"si2.svg\"><mml:msup><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">+</mml:mo><mml:mn>1</mml:mn></mml:math> for all <ce:italic>n</ce:italic>. This paper provides an overview of the known large Sidon sets in <mml:math altimg=\"si1.svg\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"double-struck\">F</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math>, and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane.","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142874013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

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

Journal of Combinatorial Theory Series A Pub Date : 2024-12-03 DOI: 10.1016/j.jcta.2024.105984

Yushuang Fan, Qinghai Zhong

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

Journal of Combinatorial Theory Series A Pub Date : 2024-11-29 DOI: 10.1016/j.jcta.2024.105979

Michael Kiermaier , Jonathan Mannaert , Alfred Wassermann

{"title":"The degree of functions in the Johnson and q-Johnson schemes","authors":"Michael Kiermaier , Jonathan Mannaert , Alfred Wassermann","doi":"10.1016/j.jcta.2024.105979","DOIUrl":"10.1016/j.jcta.2024.105979","url":null,"abstract":"<div><div>In 1982, Cameron and Liebler investigated certain <em>special sets of lines</em> in <span><math><mi>PG</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these <em>Cameron-Liebler line classes</em> 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 <em>Johnson</em> and <em>q-Johnson schemes</em>.</div><div>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 <em>designs</em> and <em>antidesigns</em> in association schemes. This leads to a natural definition of the <em>degree</em> and the <em>weights</em> 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 <em>t</em> will be proven.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 105979"},"PeriodicalIF":0.9,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

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

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

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

Journal of Combinatorial Theory Series A Pub Date : 2024-11-26 DOI: 10.1016/j.jcta.2024.105981

Yang Huang, Yuejian Peng

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

Journal of Combinatorial Theory Series A Pub Date : 2024-11-26 DOI: 10.1016/j.jcta.2024.105983

Hongxue Liang , Alessandro Montinaro

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

Journal of Combinatorial Theory Series A Pub Date : 2024-11-26 DOI: 10.1016/j.jcta.2024.105982

Byungchan Kim , Eunmi Kim

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

Journal of Combinatorial Theory Series A Pub Date : 2024-11-16 DOI: 10.1016/j.jcta.2024.105978

Takahiro Matsushita , Shun Wakatsuki

{"title":"Dominance complexes, neighborhood complexes and combinatorial Alexander duals","authors":"Takahiro Matsushita , Shun Wakatsuki","doi":"10.1016/j.jcta.2024.105978","DOIUrl":"10.1016/j.jcta.2024.105978","url":null,"abstract":"<div><div>We show that the dominance complex <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> coincides with the combinatorial Alexander dual of the neighborhood complex <span><math><mi>N</mi><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span> of the complement of <em>G</em>. Using this, we obtain a relation between the chromatic number <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em> and the homology group of <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. 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 <span><math><mi>D</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> by determining the homotopy types of the independence complex of <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105978"},"PeriodicalIF":0.9,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

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

Journal of Combinatorial Theory Series A Pub Date : 2024-11-06 DOI: 10.1016/j.jcta.2024.105968

Sam Mattheus, Geertrui Van de Voorde

{"title":"Upper bounds for the number of substructures in finite geometries from the container method","authors":"Sam Mattheus, Geertrui Van de Voorde","doi":"10.1016/j.jcta.2024.105968","DOIUrl":"10.1016/j.jcta.2024.105968","url":null,"abstract":"<div><div>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 <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mi>r</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and plane spreads in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, and caps in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. The latter result extends work due to Roche-Newton and Warren <span><span>[21]</span></span> and Bhowmick and Roche-Newton <span><span>[6]</span></span>.</div><div>Finally, we investigate caps in <em>p</em>-random subsets of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, which parallels recent work for arcs in projective planes by Bhowmick and Roche-Newton, and Roche-Newton and Warren <span><span>[6]</span></span>, <span><span>[21]</span></span>, and arcs in projective spaces by Chen, Liu, Nie and Zeng <span><span>[8]</span></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105968"},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

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

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

{"title":"The vector space generated by permutations of a trade or a design","authors":"E. Ghorbani , S. Kamali , G.B. Khosrovshahi","doi":"10.1016/j.jcta.2024.105969","DOIUrl":"10.1016/j.jcta.2024.105969","url":null,"abstract":"<div><div>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 <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> minimal trades generate the vector space of all <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> 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 <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design, its permutations can span the vector space generated by all <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> designs for sufficiently large values of <em>v</em>. In other words, any <em>t</em>-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design, or even any <em>t</em>-trade, can be expressed as a linear combination of permutations of a fixed <em>t</em>-design. This substantially extends a result by Ghodrati (2019), who proved the same result for Steiner designs.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105969"},"PeriodicalIF":0.9,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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