Journal of Combinatorial Designs最新文献_第10页

Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG ( 3 , q ) $(3,q)$ PG(3,q)室上克奈瑟图的最大茧和色度数

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-04-15 DOI: 10.1002/jcd.21940

Philipp Heering, Klaus Metsch

{"title":"Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG\u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 3\u0000 ,\u0000 q\u0000 \u0000 )\u0000 \u0000 \u0000 $(3,q)$","authors":"Philipp Heering, Klaus Metsch","doi":"10.1002/jcd.21940","DOIUrl":"10.1002/jcd.21940","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}$</annotation>\u0000 </semantics></math> be the graph whose vertices are the chambers of the finite projective 3-space <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(3,q)$</annotation>\u0000 </semantics></math>, with two vertices being adjacent if and only if the corresponding chambers are in general position. We show that a maximal independent set of vertices of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}$</annotation>\u0000 </semantics></math> contains <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>4</mn>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 <mi>q</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> ${q}^{4}+3{q}^{3}+4{q}^{2}+3q+1$</annotation>\u0000 </semantics></math>, or <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>5</mn>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 <mi>q</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $3{q}^{3}+5{q}^{2}+3q+1$</annotation>\u0000 </semantics></math>, or at most <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <msup>\u0000 <mi>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 7","pages":"388-409"},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21940","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An explicit construction for large sets of infinite dimensional q $q$ -Steiner systems 无穷维 q-Steiner 系统大集合的显式构造

IF 0.7 4区数学

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

Daniel R. Hawtin

{"title":"An explicit construction for large sets of infinite dimensional \u0000 \u0000 \u0000 q\u0000 \u0000 $q$\u0000 -Steiner systems","authors":"Daniel R. Hawtin","doi":"10.1002/jcd.21942","DOIUrl":"10.1002/jcd.21942","url":null,"abstract":"Let <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 </mrow>\u0000 <annotation> $V$</annotation>\u0000 </semantics></math> be a vector space over the finite field <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{q}$</annotation>\u0000 </semantics></math>. A <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>-Steiner system, or an <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 \u0000 <msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>V</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $S{(t,k,V)}_{q}$</annotation>\u0000 </semantics></math>, is a collection <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℬ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{{mathcal B}}}$</annotation>\u0000 </semantics></math> of <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-dimensional subspaces of <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 </mrow>\u0000 <annotation> $V$</annotation>\u0000 </semantics></math> such that every <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-dimensional subspace of <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 </mrow>\u0000 <annotation> $V$</annotation>\u0000 </semantics></math> is contained in a unique element of <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℬ</mi>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 8","pages":"413-418"},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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

Genuinely nonabelian partial difference sets 真正非阿贝尔局部差集

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-03-30 DOI: 10.1002/jcd.21938

John Polhill, James A. Davis, Ken W. Smith, Eric Swartz

{"title":"Genuinely nonabelian partial difference sets","authors":"John Polhill, James A. Davis, Ken W. Smith, Eric Swartz","doi":"10.1002/jcd.21938","DOIUrl":"10.1002/jcd.21938","url":null,"abstract":"Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 7","pages":"351-370"},"PeriodicalIF":0.7,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21938","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Doubly sequenceable groups 双序列群

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-03-30 DOI: 10.1002/jcd.21939

Mohammad Javaheri

{"title":"Doubly sequenceable groups","authors":"Mohammad Javaheri","doi":"10.1002/jcd.21939","DOIUrl":"10.1002/jcd.21939","url":null,"abstract":"Given a sequence <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 \u0000 <mo>:</mo>\u0000 \u0000 <msub>\u0000 <mi>g</mi>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>g</mi>\u0000 \u0000 <mi>m</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${bf{g}}:{g}_{0},{rm{ldots }},{g}_{m}$</annotation>\u0000 </semantics></math> in a finite group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>g</mi>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <msub>\u0000 <mn>1</mn>\u0000 \u0000 <mi>G</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${g}_{0}={1}_{G}$</annotation>\u0000 </semantics></math>, let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mover>\u0000 <mi>g</mi>\u0000 \u0000 <mo>¯</mo>\u0000 </mover>\u0000 \u0000 <mo>:</mo>\u0000 \u0000 <msub>\u0000 <mover>\u0000 <mi>g</mi>\u0000 \u0000 <mo>¯</mo>\u0000 </mover>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mover>\u0000 <mi>g</mi>\u0000 \u0000 <mo>¯</mo>\u0000 </mover>\u0000 \u0000 <mi>m</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $bar{{bf{g}}}:{bar{g}}_{0},{rm{ldots }},{bar{g}}_{m}$</annotation>\u0000 </semantics></math> be the sequence of consecutive quotients of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 </mrow>\u0000 <annotation> ${bf{g}}$</annotation>\u0000 </semantics></math> defined by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mover>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 7","pages":"371-387"},"PeriodicalIF":0.7,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140570602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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

Group divisible designs with block size 4 and group sizes 4 and 7 组块大小为 4，组块大小为 4 和 7 的可分割设计

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-03-28 DOI: 10.1002/jcd.21932

R. Julian R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe

{"title":"Group divisible designs with block size 4 and group sizes 4 and 7","authors":"R. Julian R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe","doi":"10.1002/jcd.21932","DOIUrl":"https://doi.org/10.1002/jcd.21932","url":null,"abstract":"In this paper, we consider the existence of group divisible designs (GDDs) with block size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $4$</annotation>\u0000 </semantics></math> and group sizes <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $4$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>7</mn>\u0000 </mrow>\u0000 <annotation> $7$</annotation>\u0000 </semantics></math>. We show that there exists a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $4$</annotation>\u0000 </semantics></math>-GDD of type <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>4</mn>\u0000 <mi>t</mi>\u0000 </msup>\u0000 <msup>\u0000 <mn>7</mn>\u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${4}^{t}{7}^{s}$</annotation>\u0000 </semantics></math> for all but a finite specified set of feasible values for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>,</mo>\u0000 <mi>s</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(t,s)$</annotation>\u0000 </semantics></math>.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 6","pages":"328-346"},"PeriodicalIF":0.7,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21932","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140553167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Alternating parity weak sequencing 交替奇偶校验弱排序

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-02-27 DOI: 10.1002/jcd.21936

Simone Costa, Stefano Della Fiore

{"title":"Alternating parity weak sequencing","authors":"Simone Costa, Stefano Della Fiore","doi":"10.1002/jcd.21936","DOIUrl":"10.1002/jcd.21936","url":null,"abstract":"A subset <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> $S$</annotation>\u0000 </semantics></math> of a group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mo>+</mo>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(G,+)$</annotation>\u0000 </semantics></math> is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-weakly sequenceable if there is an ordering <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mi>y</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>y</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $({y}_{1},{rm{ldots }},{y}_{k})$</annotation>\u0000 </semantics></math> of its elements such that the partial sums <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>s</mi>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>s</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>s</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${s}_{0},{s}_{1},{rm{ldots }},{s}_{k}$</annotation>\u0000 </semantics></math>, given by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>s</mi>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation> ${s}_{0}=0$</annotation>\u0000 </","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 6","pages":"308-327"},"PeriodicalIF":0.7,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139978158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 existence of 2 3 ${2}^{3}$ -decomposable super-simple ( v , 4 , 6 ) $(v,4,6)$ -BIBDs 存在23个可分解的超简单（v,4,6）$(v,4,6)$-BIBDs

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-02-22 DOI: 10.1002/jcd.21935

Huangsheng Yu, Jingyuan Chen, R. Julian R. Abel, Dianhua Wu

{"title":"The existence of \u0000 \u0000 \u0000 \u0000 2\u0000 3\u0000 \u0000 \u0000 ${2}^{3}$\u0000 -decomposable super-simple \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 v\u0000 ,\u0000 4\u0000 ,\u0000 6\u0000 \u0000 )\u0000 \u0000 \u0000 $(v,4,6)$\u0000 -BIBDs","authors":"Huangsheng Yu, Jingyuan Chen, R. Julian R. Abel, Dianhua Wu","doi":"10.1002/jcd.21935","DOIUrl":"10.1002/jcd.21935","url":null,"abstract":"A design is said to be super-simple if the intersection of any two blocks has at most two elements. A design with index <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $tlambda $</annotation>\u0000 </semantics></math> is said to be <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>λ</mi>\u0000 <mi>t</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${lambda }^{t}$</annotation>\u0000 </semantics></math>-decomposable, if its blocks can be partitioned into nonempty collections <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>≤</mo>\u0000 <mi>i</mi>\u0000 <mo>≤</mo>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $1le ile t$</annotation>\u0000 </semantics></math>, such that each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{rm{ {mathcal B} }}}_{i}$</annotation>\u0000 </semantics></math> with the point set forms a design with index <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math>. In this paper, it is proved that there exists a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mn>3</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${2}^{3}$</annotation>\u0000 </semantics></math>-decomposable super-simple <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>,</mo>\u0000 <mn>4</mn>\u0000 <mo>,</mo>\u0000 <mn>6</mn>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v,4,6)$</annotation>\u0000 </semantics></math>-BIBD (balanced incomplete block design) if and only if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>≥</mo>\u0000","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 6","pages":"297-307"},"PeriodicalIF":0.7,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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

Partition of ordered triples into uniform holey ordered designs 将有序三元组划分为均匀有序的洞穴设计

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-02-19 DOI: 10.1002/jcd.21933

Yuli Tan, Junling Zhou

{"title":"Partition of ordered triples into uniform holey ordered designs","authors":"Yuli Tan, Junling Zhou","doi":"10.1002/jcd.21933","DOIUrl":"10.1002/jcd.21933","url":null,"abstract":"A large set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>LOD</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{LOD}(v)$</annotation>\u0000 </semantics></math> is a partition of all ordered triples of a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>-set into <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $v-2$</annotation>\u0000 </semantics></math> disjoint ordered designs of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>. In this paper, we generalize the large set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>LOD</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{LOD}(v)$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $v=gt$</annotation>\u0000 </semantics></math> to the notion of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>POT</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{POT}({g}^{t})$</annotation>\u0000 </semantics></math>, representing a partition of all ordered triples of a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $gt$</annotation>\u0000 </semantics></math>-set into disjoint uniform holely ordered designs <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>HOD</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{HOD}({g}^{t})$</annotation>\u0000 </semantics></math>s. We show that a <math>\u0000 <semantics","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 5","pages":"274-293"},"PeriodicalIF":0.7,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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

Geometric dual and sum-rank minimal codes 几何对偶码与和秩最小码

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-02-14 DOI: 10.1002/jcd.21934

Martino Borello, Ferdinando Zullo

引用次数: 0

Constructing MRD codes by switching 通过切换构建 MRD 代码

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2024-02-08 DOI: 10.1002/jcd.21931

Minjia Shi, Denis S. Krotov, Ferruh Özbudak

{"title":"Constructing MRD codes by switching","authors":"Minjia Shi, Denis S. Krotov, Ferruh Özbudak","doi":"10.1002/jcd.21931","DOIUrl":"10.1002/jcd.21931","url":null,"abstract":"Maximum rank-distance (MRD) codes are (not necessarily linear) maximum codes in the rank-distance metric space on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-by-<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> matrices over a finite field <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{q}$</annotation>\u0000 </semantics></math>. They are diameter perfect and have the cardinality <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mi>d</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${q}^{m(n-d+1)}$</annotation>\u0000 </semantics></math> if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>≥</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $mge n$</annotation>\u0000 </semantics></math>. We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> if the other parameters (<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $n,,q$</annotation>\u0000 </semantics></math>, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"32 5","pages":"219-237"},"PeriodicalIF":0.7,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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