An explicit construction for large sets of infinite dimensional q $q$ -Steiner systems

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2024-04-15 DOI:10.1002/jcd.21942

Daniel R. Hawtin

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A <math>\n \n <semantics>\n <mrow>\n <mi>q</mi>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math>-Steiner system, or an <math>\n \n <semantics>\n <mrow>\n <mi>S</mi>\n \n <msub>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>t</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>V</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> $S{(t,k,V)}_{q}$</annotation>\n </semantics></math>, is a collection <math>\n \n <semantics>\n <mrow>\n <mi>ℬ</mi>\n </mrow>\n <annotation> ${\\rm{{\\mathcal B}}}$</annotation>\n </semantics></math> of <math>\n \n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-dimensional subspaces of <math>\n \n <semantics>\n <mrow>\n <mi>V</mi>\n </mrow>\n <annotation> $V$</annotation>\n </semantics></math> such that every <math>\n \n <semantics>\n <mrow>\n <mi>t</mi>\n </mrow>\n <annotation> $t$</annotation>\n </semantics></math>-dimensional subspace of <math>\n \n <semantics>\n <mrow>\n <mi>V</mi>\n </mrow>\n <annotation> $V$</annotation>\n </semantics></math> is contained in a unique element of <math>\n \n <semantics>\n <mrow>\n <mi>ℬ</mi>\n </mrow>\n <annotation> ${\\rm{{\\mathcal B}}}$</annotation>\n </semantics></math>. A large set of <math>\n \n <semantics>\n <mrow>\n <mi>q</mi>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math>-Steiner systems, or an <math>\n \n <semantics>\n <mrow>\n <mi>L</mi>\n \n <mi>S</mi>\n \n <msub>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>t</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>V</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> $LS{(t,k,V)}_{q}$</annotation>\n </semantics></math>, is a partition of the <math>\n \n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-dimensional subspaces of <math>\n \n <semantics>\n <mrow>\n <mi>V</mi>\n </mrow>\n <annotation> $V$</annotation>\n </semantics></math> into <math>\n \n <semantics>\n <mrow>\n <mi>S</mi>\n \n <msub>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>t</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>V</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> $S{(t,k,V)}_{q}$</annotation>\n </semantics></math> systems. In the case that <math>\n \n <semantics>\n <mrow>\n <mi>V</mi>\n </mrow>\n <annotation> $V$</annotation>\n </semantics></math> has infinite dimension, the existence of an <math>\n \n <semantics>\n <mrow>\n <mi>L</mi>\n \n <mi>S</mi>\n \n <msub>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>t</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>V</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> $LS{(t,k,V)}_{q}$</annotation>\n </semantics></math> for all finite <math>\n \n <semantics>\n <mrow>\n <mi>t</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n </mrow>\n <annotation> $t,k$</annotation>\n </semantics></math> with <math>\n \n <semantics>\n <mrow>\n <mn>1</mn>\n \n <mo><</mo>\n \n <mi>t</mi>\n \n <mo><</mo>\n \n <mi>k</mi>\n </mrow>\n <annotation> $1\\lt t\\lt k$</annotation>\n </semantics></math> was shown abstractly by Cameron in 1995. This paper provides an explicit construction of an <math>\n \n <semantics>\n <mrow>\n <mi>L</mi>\n \n <mi>S</mi>\n \n <msub>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>t</mi>\n \n <mo>,</mo>\n \n <mi>t</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <mi>V</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> $LS{(t,t+1,V)}_{q}$</annotation>\n </semantics></math> for all prime powers <math>\n \n <semantics>\n <mrow>\n <mi>q</mi>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math>, all positive integers <math>\n \n <semantics>\n <mrow>\n <mi>t</mi>\n </mrow>\n <annotation> $t$</annotation>\n </semantics></math>, and where <math>\n \n <semantics>\n <mrow>\n <mi>V</mi>\n </mrow>\n <annotation> $V$</annotation>\n </semantics></math> has countably infinite dimension.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"413-418"},"PeriodicalIF":0.5000,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21942","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $V$ be a vector space over the finite field $F_{q}$ . A $q$ -Steiner system, or an $S {(t, k, V)}_{q}$ , is a collection $ℬ$ of $k$ -dimensional subspaces of $V$ such that every $t$ -dimensional subspace of $V$ is contained in a unique element of $ℬ$ . A large set of $q$ -Steiner systems, or an $L S {(t, k, V)}_{q}$ , is a partition of the $k$ -dimensional subspaces of $V$ into $S {(t, k, V)}_{q}$ systems. In the case that $V$ has infinite dimension, the existence of an $L S {(t, k, V)}_{q}$ for all finite $t, k$ with $1 < t < k$ was shown abstractly by Cameron in 1995. This paper provides an explicit construction of an $L S {(t, t + 1, V)}_{q}$ for all prime powers $q$ , all positive integers $t$ , and where $V$ has countably infinite dimension.

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无穷维 q-Steiner 系统大集合的显式构造

设 V$V$ 是有限域 Fq${{\mathbb{F}}}_{q}$ 上的向量空间。一个 q$q$-Steiner 系统或一个 S(t,k,V)q$S{(t,k,V)}_{q}$ 是ℬ$\{rm{ {\mathcal B}}$ 的集合。V$V$ 的 k$k$ 维子空间的集合，使得 V$V$ 的每个 t$t$ 维子空间都包含在ℬ${rm{ {\mathcal B}}$ 的唯一元素中。｝}}$.一大组 q$q$-Steiner 系统或 LS(t,k,V)q$LS{(t,k,V)}_{q}$ 是将 V$V$ 的 k$k$ 维子空间划分为 S(t,k,V)q$S{(t,k,V)}_{q}$ 系统。在 V$V$ 具有无限维的情况下，卡梅伦在 1995 年抽象地证明了对于所有有限的 t,kt,k$，1<t<k$1\lt t\lt k$，LS(t,k,V)q$LS{(t,k,V)}_{q}$的存在。本文明确地构造了 LS(t,t+1,V)q$LS{(t,t+1,V)}_{q}$ ，适用于所有素数幂 q$q$，所有正整数 t$t$，且 V$V$ 具有可数无限维。

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.