完全还原的超简单 ( v , 4 , 4 ) $(v,4,4)$ -BIBD 及相关恒权码

IF 0.5 4区数学 Q3 MATHEMATICS Journal of Combinatorial Designs Pub Date : 2024-10-10 DOI:10.1002/jcd.21958

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

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A design with index <math>\n <semantics>\n <mrow>\n <mi>λ</mi>\n </mrow>\n <annotation> $\\lambda $</annotation>\n </semantics></math> is said to be completely reducible, if its blocks can be partitioned into nonempty collections <math>\n <semantics>\n <mrow>\n <msub>\n <mi>B</mi>\n \n <mi>i</mi>\n </msub>\n \n <mo>,</mo>\n \n <mn>1</mn>\n \n <mo>≤</mo>\n \n <mi>i</mi>\n \n <mo>≤</mo>\n \n <mi>λ</mi>\n </mrow>\n <annotation> ${{\\mathscr{B}}}_{i},1\\le i\\le \\lambda $</annotation>\n </semantics></math>, such that each <math>\n <semantics>\n <mrow>\n <msub>\n <mi>B</mi>\n \n <mi>i</mi>\n </msub>\n </mrow>\n <annotation> ${{\\mathscr{B}}}_{i}$</annotation>\n </semantics></math> together with the point set forms a design with index unity. In this paper, it is proved that there exists a completely reducible super-simple (CRSS) <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mn>4</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n <annotation> $(v,4,4)$</annotation>\n </semantics></math> balanced incomplete block design (<math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mn>4</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n <annotation> $(v,4,4)$</annotation>\n </semantics></math>-BIBD for short) if and only if <math>\n <semantics>\n <mrow>\n <mi>v</mi>\n \n <mo>≥</mo>\n \n <mn>13</mn>\n </mrow>\n <annotation> $v\\ge 13$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>v</mi>\n \n <mo>≡</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $v\\equiv 1$</annotation>\n </semantics></math> or <math>\n <semantics>\n <mrow>\n <mn>4</mn>\n <mspace></mspace>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>mod</mi>\n <mspace></mspace>\n \n <mn>12</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $4\\,(\\mathrm{mod}\\,12)$</annotation>\n </semantics></math>. A <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math>-ary constant weight code (CWC) of length <math>\n <semantics>\n <mrow>\n <mi>v</mi>\n </mrow>\n <annotation> $v$</annotation>\n </semantics></math> with weight <math>\n <semantics>\n <mrow>\n <mi>w</mi>\n </mrow>\n <annotation> $w$</annotation>\n </semantics></math> and distance <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math> is denoted as a <math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mi>d</mi>\n \n <mo>,</mo>\n \n <mi>w</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> ${(v,d,w)}_{q}$</annotation>\n </semantics></math> code. The maximum size of a <math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mi>d</mi>\n \n <mo>,</mo>\n \n <mi>w</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> ${(v,d,w)}_{q}$</annotation>\n </semantics></math> code is denoted as <math>\n <semantics>\n <mrow>\n <msub>\n <mi>A</mi>\n \n <mi>q</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mi>d</mi>\n \n <mo>,</mo>\n \n <mi>w</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${A}_{q}(v,d,w)$</annotation>\n </semantics></math>, and the <math>\n <semantics>\n <mrow>\n <msub>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mi>d</mi>\n \n <mo>,</mo>\n \n <mi>w</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mi>q</mi>\n </msub>\n </mrow>\n <annotation> ${(v,d,w)}_{q}$</annotation>\n </semantics></math> codes achieving this size are called optimal. CRSS designs with index <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $q-1$</annotation>\n </semantics></math> are closely related to <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math>-ary CWCs. By using the results of CRSS <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mn>4</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n <annotation> $(v,4,4)$</annotation>\n </semantics></math>-BIBDs, <math>\n <semantics>\n <mrow>\n <msub>\n <mi>A</mi>\n \n <mn>5</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>v</mi>\n \n <mo>,</mo>\n \n <mn>6</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${A}_{5}(v,6,4)$</annotation>\n </semantics></math>s are determined for all <math>\n <semantics>\n <mrow>\n <mi>v</mi>\n \n <mo>≡</mo>\n \n <mn>0</mn>\n \n <mo>,</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>3</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n <mspace></mspace>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>mod</mi>\n <mspace></mspace>\n \n <mn>12</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mi>v</mi>\n \n <mo>≥</mo>\n \n <mn>12</mn>\n </mrow>\n <annotation> $v\\equiv 0,1,3,4\\,(\\mathrm{mod}\\,12),v\\ge 12$</annotation>\n </semantics></math>.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 1","pages":"27-36"},"PeriodicalIF":0.5000,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Completely reducible super-simple \\n \\n \\n (\\n \\n v\\n ,\\n 4\\n ,\\n 4\\n \\n )\\n \\n $(v,4,4)$\\n -BIBDs and related constant weight codes\",\"authors\":\"Jingyuan Chen, Huangsheng Yu, R. Julian R. Abel, Dianhua Wu\",\"doi\":\"10.1002/jcd.21958\",\"DOIUrl\":null,\"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>\\n <semantics>\\n <mrow>\\n <mi>λ</mi>\\n </mrow>\\n <annotation> $\\\\lambda $</annotation>\\n </semantics></math> is said to be completely reducible, if its blocks can be partitioned into nonempty collections <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>B</mi>\\n \\n <mi>i</mi>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <mn>1</mn>\\n \\n <mo>≤</mo>\\n \\n <mi>i</mi>\\n \\n <mo>≤</mo>\\n \\n <mi>λ</mi>\\n </mrow>\\n <annotation> ${{\\\\mathscr{B}}}_{i},1\\\\le i\\\\le \\\\lambda $</annotation>\\n </semantics></math>, such that each <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>B</mi>\\n \\n <mi>i</mi>\\n </msub>\\n </mrow>\\n <annotation> ${{\\\\mathscr{B}}}_{i}$</annotation>\\n </semantics></math> together with the point set forms a design with index unity. In this paper, it is proved that there exists a completely reducible super-simple (CRSS) <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n <annotation> $(v,4,4)$</annotation>\\n </semantics></math> balanced incomplete block design (<math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n <annotation> $(v,4,4)$</annotation>\\n </semantics></math>-BIBD for short) if and only if <math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>≥</mo>\\n \\n <mn>13</mn>\\n </mrow>\\n <annotation> $v\\\\ge 13$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>≡</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $v\\\\equiv 1$</annotation>\\n </semantics></math> or <math>\\n <semantics>\\n <mrow>\\n <mn>4</mn>\\n <mspace></mspace>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>mod</mi>\\n <mspace></mspace>\\n \\n <mn>12</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $4\\\\,(\\\\mathrm{mod}\\\\,12)$</annotation>\\n </semantics></math>. A <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n </mrow>\\n <annotation> $q$</annotation>\\n </semantics></math>-ary constant weight code (CWC) of length <math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n </mrow>\\n <annotation> $v$</annotation>\\n </semantics></math> with weight <math>\\n <semantics>\\n <mrow>\\n <mi>w</mi>\\n </mrow>\\n <annotation> $w$</annotation>\\n </semantics></math> and distance <math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n </mrow>\\n <annotation> $d$</annotation>\\n </semantics></math> is denoted as a <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>,</mo>\\n \\n <mi>d</mi>\\n \\n <mo>,</mo>\\n \\n <mi>w</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mi>q</mi>\\n </msub>\\n </mrow>\\n <annotation> ${(v,d,w)}_{q}$</annotation>\\n </semantics></math> code. The maximum size of a <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>,</mo>\\n \\n <mi>d</mi>\\n \\n <mo>,</mo>\\n \\n <mi>w</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mi>q</mi>\\n </msub>\\n </mrow>\\n <annotation> ${(v,d,w)}_{q}$</annotation>\\n </semantics></math> code is denoted as <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>A</mi>\\n \\n <mi>q</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>,</mo>\\n \\n <mi>d</mi>\\n \\n <mo>,</mo>\\n \\n <mi>w</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> ${A}_{q}(v,d,w)$</annotation>\\n </semantics></math>, and the <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>,</mo>\\n \\n <mi>d</mi>\\n \\n <mo>,</mo>\\n \\n <mi>w</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mi>q</mi>\\n </msub>\\n </mrow>\\n <annotation> ${(v,d,w)}_{q}$</annotation>\\n </semantics></math> codes achieving this size are called optimal. CRSS designs with index <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $q-1$</annotation>\\n </semantics></math> are closely related to <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n </mrow>\\n <annotation> $q$</annotation>\\n </semantics></math>-ary CWCs. By using the results of CRSS <math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n <annotation> $(v,4,4)$</annotation>\\n </semantics></math>-BIBDs, <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>A</mi>\\n \\n <mn>5</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>,</mo>\\n \\n <mn>6</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> ${A}_{5}(v,6,4)$</annotation>\\n </semantics></math>s are determined for all <math>\\n <semantics>\\n <mrow>\\n <mi>v</mi>\\n \\n <mo>≡</mo>\\n \\n <mn>0</mn>\\n \\n <mo>,</mo>\\n \\n <mn>1</mn>\\n \\n <mo>,</mo>\\n \\n <mn>3</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n <mspace></mspace>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>mod</mi>\\n <mspace></mspace>\\n \\n <mn>12</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>,</mo>\\n \\n <mi>v</mi>\\n \\n <mo>≥</mo>\\n \\n <mn>12</mn>\\n </mrow>\\n <annotation> $v\\\\equiv 0,1,3,4\\\\,(\\\\mathrm{mod}\\\\,12),v\\\\ge 12$</annotation>\\n </semantics></math>.\",\"PeriodicalId\":15389,\"journal\":{\"name\":\"Journal of Combinatorial Designs\",\"volume\":\"33 1\",\"pages\":\"27-36\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-10-10\",\"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.21958\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21958","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

摘要

( v , d , w ) q ${(v,d,w)}_{q}$ 编码的最大大小记为 A q ( v , d , w ) ${A}_{q}(v,d,w)$ ，达到这一大小的 ( v , d , w ) q ${(v,d,w)}_{q}$ 编码称为最优编码。索引为 q - 1 $q-1$ 的 CRSS 设计与 q $q$ -ary CWC 密切相关。利用 CRSS ( v , 4 , 4 ) $(v,4,4)$ -BIBDs 的结果，可以确定 A 5 ( v , 6 , 4 ) ${A}_{5}(v,6,4)$ s 适用于所有 v ≡ 0 , 1 , 3 , 4 ( mod 12 ) , v ≥ 12 $v\equiv 0,1,3,4\,(\mathrm{mod}\,12),v\ge 12$ .

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Completely reducible super-simple ( v , 4 , 4 ) $(v,4,4)$ -BIBDs and related constant weight codes

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A design with index $λ$ is said to be completely reducible, if its blocks can be partitioned into nonempty collections $B_{i}, 1 \leq i \leq λ$ , such that each $B_{i}$ together with the point set forms a design with index unity. In this paper, it is proved that there exists a completely reducible super-simple (CRSS) $(v, 4, 4)$ balanced incomplete block design ( $(v, 4, 4)$ -BIBD for short) if and only if $v \geq 13$ and $v \equiv 1$ or $4 (\mod 12)$ . A $q$ -ary constant weight code (CWC) of length $v$ with weight $w$ and distance $d$ is denoted as a ${(v, d, w)}_{q}$ code. The maximum size of a ${(v, d, w)}_{q}$ code is denoted as $A_{q} (v, d, w)$ , and the ${(v, d, w)}_{q}$ codes achieving this size are called optimal. CRSS designs with index $q - 1$ are closely related to $q$ -ary CWCs. By using the results of CRSS $(v, 4, 4)$ -BIBDs, $A_{5} (v, 6, 4)$ s are determined for all $v \equiv 0, 1, 3, 4 (\mod 12), v \geq 12$ .

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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.