In the present paper, we give proofs of the existence of a 3-design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18.
{"title":"Exceptional designs in some extended quadratic residue codes","authors":"Reina Ishikawa","doi":"10.1002/jcd.21907","DOIUrl":"https://doi.org/10.1002/jcd.21907","url":null,"abstract":"<p>In the present paper, we give proofs of the existence of a 3-design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"496-510"},"PeriodicalIF":0.7,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50152059","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}
We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes of such objects. Totally symmetric quasigroups of orders up to and including 16 that are (respectively) medial, idempotent, and unipotent are also enumerated.
{"title":"Totally symmetric quasigroups of order 16","authors":"Hy Ginsberg","doi":"10.1002/jcd.21910","DOIUrl":"https://doi.org/10.1002/jcd.21910","url":null,"abstract":"<p>We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes of such objects. Totally symmetric quasigroups of orders up to and including 16 that are (respectively) medial, idempotent, and unipotent are also enumerated.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"531-542"},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50131265","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}
Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used to obtain the number of isomorphism classes for order 21: 14,796,207,517,873,771.
{"title":"Enumerating Steiner triple systems","authors":"Daniel Heinlein, Patric R. J. Östergård","doi":"10.1002/jcd.21906","DOIUrl":"https://doi.org/10.1002/jcd.21906","url":null,"abstract":"<p>Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used to obtain the number of isomorphism classes for order 21: 14,796,207,517,873,771.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"479-495"},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21906","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50140412","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}
A Latin square of order is an matrix of symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power let denote the finite field of order . A quadratic Latin square is a Latin square defined by
{"title":"Cycles of quadratic Latin squares and antiperfect 1-factorisations","authors":"Jack Allsop","doi":"10.1002/jcd.21905","DOIUrl":"https://doi.org/10.1002/jcd.21905","url":null,"abstract":"<p>A Latin square of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $ntimes n$</annotation>\u0000 </semantics></math> matrix of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math> let <math>\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> denote the finite field of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>. A quadratic Latin square is a Latin square <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 \u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>b</mi>\u0000 </mrow>\u0000 \u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal L} }}[a,b]$</annotation>\u0000 </semantics></math> defined by\u0000\u0000 </p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"447-475"},"PeriodicalIF":0.7,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21905","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50127978","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}
<p>A line coloring of <math> <semantics> <mrow> <mtext>PG</mtext> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $text{PG}(n,q)$</annotation> </semantics></math>, the <math> <semantics> <mrow> <mi>n</mi> </mrow> <annotation> $n$</annotation> </semantics></math>-dimensional projective space over GF<math> <semantics> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> <annotation> $(q)$</annotation> </semantics></math>, is an assignment of colors to all lines of <math> <semantics> <mrow> <mtext>PG</mtext> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $text{PG}(n,q)$</annotation> </semantics></math> so that any two lines with the same color do not intersect. The chromatic index of <math> <semantics> <mrow> <mtext>PG</mtext> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $text{PG}(n,q)$</annotation> </semantics></math>, denoted by <math> <semantics> <mrow> <mi>χ</mi> <mo>′</mo> <mrow> <mo>(</mo> <mrow> <mtext>PG</mtext> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</
We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.
{"title":"Existence of small ordered orthogonal arrays","authors":"Kai-Uwe Schmidt, Charlene Weiß","doi":"10.1002/jcd.21903","DOIUrl":"https://doi.org/10.1002/jcd.21903","url":null,"abstract":"<p>We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"422-431"},"PeriodicalIF":0.7,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21903","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50152419","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}
<p>A <i>single-change covering design</i> (SCCD) is a <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math>-set <math> <semantics> <mrow> <mi>X</mi> </mrow> <annotation> $X$</annotation> </semantics></math> and an ordered list <math> <semantics> <mrow> <mi>ℒ</mi> </mrow> <annotation> ${rm{ {mathcal L} }}$</annotation> </semantics></math> of <math> <semantics> <mrow> <mi>b</mi> </mrow> <annotation> $b$</annotation> </semantics></math> blocks of size <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math> where every pair from <math> <semantics> <mrow> <mi>X</mi> </mrow> <annotation> $X$</annotation> </semantics></math> must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given <math> <semantics> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> </mrow> <annotation> $v,k$</annotation> </semantics></math>. In this paper, we use a new recursive construction to solve the existence of circular SCCD(<math> <semantics> <mrow> <mi>v</mi> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mi>b</mi> </mrow> <annotation> $v,4,b$</annotation> </semantics></math>) for all <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math> and three residue classes of circular SCCD(<math> <semantics> <mrow> <mi>v</mi> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mi>b</mi> </mrow> <annotation> $v,5,b$</annotation> </semantics
{"title":"Linear and circular single-change covering designs revisited","authors":"Amanda Chafee, Brett Stevens","doi":"10.1002/jcd.21885","DOIUrl":"https://doi.org/10.1002/jcd.21885","url":null,"abstract":"<p>A <i>single-change covering design</i> (SCCD) is a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>-set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> and an ordered list <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℒ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal L} }}$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $b$</annotation>\u0000 </semantics></math> blocks of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> where every pair from <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>,</mo>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $v,k$</annotation>\u0000 </semantics></math>. In this paper, we use a new recursive construction to solve the existence of circular SCCD(<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>,</mo>\u0000 <mn>4</mn>\u0000 <mo>,</mo>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $v,4,b$</annotation>\u0000 </semantics></math>) for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math> and three residue classes of circular SCCD(<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>,</mo>\u0000 <mn>5</mn>\u0000 <mo>,</mo>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $v,5,b$</annotation>\u0000 </semantics","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"405-421"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21885","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50116119","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}
<p>Partial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self-dual association schemes. Then using the characterizations of self-dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self-dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2-designs. In particular, for a primitive self-dual symmetric association scheme <math> <semantics> <mrow> <mi>X</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>,</mo> <msub> <mrow> <mo>{</mo> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>}</mo> </mrow> <mrow> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> ${mathscr{X}}=(X,{{{R}_{i}}}_{0le ile 3})$</annotation> </semantics></math> of rank 4, if <math> <semantics> <mrow> <mo>∣</mo> <mi>X</mi> <mo>∣</mo> </mrow> <annotation> $| X| $</annotation> </semantics></math> is a power of 3 and each of <math> <semantics> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> </mrow> <annotation> $
局部几何设计可以由关联方案的基本关系来构造。Nowak等人（2016）根据某些Hamming方案的融合方案构建了一个无限族的局部几何设计。Xu（2023）给出了一种从关联方案创建局部几何设计的通用方法。在本文中，我们继续徐（2023）的研究。我们将首先研究自对偶关联方案的性质和特征。然后利用自对偶关联方案的特征和交换关联方案的表示理论（特征表），我们得到了秩为4的自对偶（对称或非对称）关联方案的刻画和分类，这些方案产生尽可能多的非平凡部分几何设计或2-设计。特别地，对于基元自对偶对称关联方案X＝（X，｛R i｝0≤i≤3）$｛mathscr｛X｝｝＝（X{{{R}_{i} ｛0le ile 3｝）$，如果｜X｜$|X|$是3的幂，并且R中的每一个为1${R}_{1} $，R 2${R}_{2} $和R0õR3${R}_｛0｝杯{R}_{3} $引入了部分几何设计，则我们将证明X$｛mathscr｛X｝｝$代数同构于Hamming方案H的一个融合方案（d，3）$H（d，三）$对于某个奇数d$d$。
{"title":"Self-dual association schemes, fusions of Hamming schemes, and partial geometric designs","authors":"Bangteng Xu","doi":"10.1002/jcd.21889","DOIUrl":"https://doi.org/10.1002/jcd.21889","url":null,"abstract":"<p>Partial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self-dual association schemes. Then using the characterizations of self-dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self-dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2-designs. In particular, for a primitive self-dual symmetric association scheme <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>i</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${mathscr{X}}=(X,{{{R}_{i}}}_{0le ile 3})$</annotation>\u0000 </semantics></math> of rank 4, if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>X</mi>\u0000 \u0000 <mo>∣</mo>\u0000 </mrow>\u0000 <annotation> $| X| $</annotation>\u0000 </semantics></math> is a power of 3 and each of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"373-399"},"PeriodicalIF":0.7,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50120463","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}
A decomposition of a graph is primitive if no proper, nontrivial subset of is a decomposition of an induced subgraph of . An unresolved question posed by Asplund et al. in a recent publication involves the existence of primitive decompositions of cocktail party graphs into cycles of length 4, which is resolved by this paper.
{"title":"Primitive \u0000 \u0000 \u0000 \u0000 C\u0000 4\u0000 \u0000 \u0000 ${C}_{4}$\u0000 -decompositions of \u0000 \u0000 \u0000 \u0000 K\u0000 n\u0000 \u0000 −\u0000 I\u0000 \u0000 ${K}_{n}-I$","authors":"Michael W. Schroeder","doi":"10.1002/jcd.21887","DOIUrl":"https://doi.org/10.1002/jcd.21887","url":null,"abstract":"<p>A decomposition <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{C}}$</annotation>\u0000 </semantics></math> of a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is primitive if no proper, nontrivial subset of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{C}}$</annotation>\u0000 </semantics></math> is a decomposition of an induced subgraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. An unresolved question posed by Asplund et al. in a recent publication involves the existence of primitive decompositions of cocktail party graphs into cycles of length 4, which is resolved by this paper.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"368-372"},"PeriodicalIF":0.7,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50122041","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}
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