{"title":"On common energies and sumsets","authors":"Shkredov I.D.","doi":"10.1016/j.jcta.2025.106026","DOIUrl":"10.1016/j.jcta.2025.106026","url":null,"abstract":"<div><div>We obtain a polynomial criterion for a set to have a small doubling in terms of the common energy of its subsets.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106026"},"PeriodicalIF":0.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474228","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}
{"title":"Structure of Terwilliger algebras of quasi-thin association schemes","authors":"Zhenxian Chen , Changchang Xi","doi":"10.1016/j.jcta.2025.106024","DOIUrl":"10.1016/j.jcta.2025.106024","url":null,"abstract":"<div><div>We show that the Terwilliger algebra of a quasi-thin association scheme over a field is always a quasi-hereditary cellular algebra in the sense of Cline-Parshall-Scott and of Graham-Lehrer, respectively, and that the basic algebra of the Terwilliger algebra is the dual extension of a star with all arrows pointing to its center if the field has characteristic 2. Thus many homological and representation-theoretic properties of these Terwilliger algebras can be determined completely. For example, the Nakayama conjecture holds true for Terwilliger algebras of quasi-thin association schemes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106024"},"PeriodicalIF":0.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480678","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}
{"title":"A symmetry on weakly increasing trees and multiset Schett polynomials","authors":"Zhicong Lin , Jun Ma","doi":"10.1016/j.jcta.2025.106010","DOIUrl":"10.1016/j.jcta.2025.106010","url":null,"abstract":"<div><div>By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma–Mansour–Wang–Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin–Ma–Ma–Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of “even-degree nodes on odd levels” and “odd-degree nodes” on weakly increasing trees is found, extending the Schett polynomials, a generalization of the Jacobi elliptic functions introduced by Schett, to multisets. A combinatorial proof and an algebraic proof of this symmetry are provided, as well as several relevant interesting consequences. Moreover, via introducing a group action on trees, we prove the partial <em>γ</em>-positivity of the multiset Schett polynomials, a result which implies both the symmetry and the unimodality of these polynomials.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106010"},"PeriodicalIF":0.9,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143103357","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}
{"title":"On recursive constructions for 2-designs over finite fields","authors":"Xiaoran Wang, Junling Zhou","doi":"10.1016/j.jcta.2025.106006","DOIUrl":"10.1016/j.jcta.2025.106006","url":null,"abstract":"<div><div>This paper concentrates on recursive constructions for 2-designs over finite fields. In 1998, Itoh presented a powerful recursive construction: for certain index <em>λ</em>, if there exists a Singer cycle invariant 2-<span><math><msub><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mn>3</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> design, then there also exists an SL<span><math><mo>(</mo><mi>m</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>)</mo></math></span> invariant 2-<span><math><msub><mrow><mo>(</mo><mi>m</mi><mi>l</mi><mo>,</mo><mn>3</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> design for all integers <span><math><mi>m</mi><mo>≥</mo><mn>3</mn></math></span>. We investigate the <span><math><mrow><mi>GL</mi></mrow><mo>(</mo><mi>m</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>)</mo></math></span>-incidence matrix between 2-subspaces and <em>k</em>-subspaces of <span><math><mi>GF</mi><mspace></mspace><msup><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mi>m</mi><mi>l</mi></mrow></msup></math></span> with <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> in this work. As a generalization of Itoh's construction, the important case of <span><math><mi>m</mi><mo>=</mo><mn>2</mn></math></span> is supplemented and a doubling construction is established for 2-<span><math><msub><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mn>3</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> designs over finite fields. As a further generalization, a product construction is developed for <em>q</em>-analogs of group divisible designs (<em>q</em>-GDDs). For general block dimension <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, several new infinite families of <em>q</em>-GDDs are constructed. As applications, plenty of new infinite families of 2-designs over finite fields are constructed.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106006"},"PeriodicalIF":0.9,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143103356","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}
{"title":"On a conjecture concerning the r-Euler-Mahonian statistic on permutations","authors":"Kaimei Huang , Zhicong Lin , Sherry H.F. Yan","doi":"10.1016/j.jcta.2025.106008","DOIUrl":"10.1016/j.jcta.2025.106008","url":null,"abstract":"<div><div>A pair <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> of permutation statistics is said to be <em>r</em>-Euler-Mahonian if <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> and (rdes, rmaj) are equidistributed over the set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all permutations of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>, where rdes denotes the <em>r</em>-descent number and rmaj denotes the <em>r</em>-major index introduced by Rawlings. The main objective of this paper is to prove that <span><math><mo>(</mo><msub><mrow><mi>exc</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>den</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo></math></span> and (rdes, rmaj) are equidistributed over <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, thereby confirming a recent conjecture posed by Liu. When <span><math><mi>r</mi><mo>=</mo><mn>1</mn></math></span>, the result recovers the equidistribution of <span><math><mo>(</mo><mrow><mi>des</mi></mrow><mo>,</mo><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>den</mi></mrow><mo>)</mo></math></span>, which was first conjectured by Denert and proved by Foata and Zeilberger.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106008"},"PeriodicalIF":0.9,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095725","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}
Jason P. Bell, Sean Monahan, Matthew Satriano, Karen Situ, Zheng Xie
{"title":"There are no good infinite families of toric codes","authors":"Jason P. Bell, Sean Monahan, Matthew Satriano, Karen Situ, Zheng Xie","doi":"10.1016/j.jcta.2025.106009","DOIUrl":"10.1016/j.jcta.2025.106009","url":null,"abstract":"<div><div>Soprunov and Soprunova posed a question on the existence of infinite families of toric codes that are “good” in a precise sense. We prove that such good families do not exist by proving a more general Szemerédi-type result: for all <span><math><mi>c</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and all positive integers <em>N</em>, subsets of density at least <em>c</em> in <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> contain hypercubes of arbitrarily large dimension as <em>n</em> grows.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106009"},"PeriodicalIF":0.9,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143103292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Unique representations of integers by linear forms","authors":"Sándor Z. Kiss , Csaba Sándor","doi":"10.1016/j.jcta.2025.106007","DOIUrl":"10.1016/j.jcta.2025.106007","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> be an integer and let <em>A</em> be a set of nonnegative integers. For a <em>k</em>-tuple of positive integers <span><math><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder><mo>=</mo><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> with <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, we define the additive representation function <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mo>,</mo><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>|</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>:</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>}</mo><mo>|</mo></math></span>. For <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, Moser constructed a set <em>A</em> of nonnegative integers such that <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mo>,</mo><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> holds for every nonnegative integer <em>n</em>. In this paper we characterize all the <em>k</em>-tuples <span><math><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder></math></span> and the sets <em>A</em> of nonnegative integers with <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi><mo>,</mo><munder><mrow><mi>λ</mi></mrow><mo>_</mo></munder></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> for every integer <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"213 ","pages":"Article 106007"},"PeriodicalIF":0.9,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Full weight spectrum one-orbit cyclic subspace codes","authors":"Chiara Castello, Olga Polverino, Ferdinando Zullo","doi":"10.1016/j.jcta.2024.106005","DOIUrl":"10.1016/j.jcta.2024.106005","url":null,"abstract":"<div><div>For a linear Hamming metric code of length <em>n</em> over a finite field, the number of distinct weights of its codewords is at most <em>n</em>. 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 <em>full weight spectrum</em> in this context. Utilizing number-theoretical results and combinatorial arguments, we provide a complete classification of full weight spectrum one-orbit cyclic subspace codes.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106005"},"PeriodicalIF":0.9,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143095723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Contributions to Ma's conjecture concerning abelian difference sets with multiplier −1 (I)","authors":"Yasutsugu Fujita , Maohua Le","doi":"10.1016/j.jcta.2024.106004","DOIUrl":"10.1016/j.jcta.2024.106004","url":null,"abstract":"<div><div>Let <span><math><mi>N</mi></math></span>, <span><math><mi>P</mi></math></span> 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 <span><span>[14]</span></span> conjectured that the equation <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> <span><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></span>, <span><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></span> has only one solution <span><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></span>. 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 <span><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></span> is a solution of <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> with <span><math><mi>m</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span>, then there exist an odd positive integer <em>g</em> and a positive integer <em>t</em> which make <span><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></span> and <span><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></span>, where <span><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></span> for any integer <em>r</em>, <span><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","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106004"},"PeriodicalIF":0.9,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143100974","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}
{"title":"Sidon sets, thin sets, and the nonlinearity of vectorial Boolean functions","authors":"Gábor P. Nagy","doi":"10.1016/j.jcta.2024.106001","DOIUrl":"10.1016/j.jcta.2024.106001","url":null,"abstract":"<div><div>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 <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, there should be Sidon sets of size at least <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span> for all <em>n</em>. This paper provides an overview of the known large Sidon sets in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"212 ","pages":"Article 106001"},"PeriodicalIF":0.9,"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}
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