{"title":"On the equivalence of certain quasi-Hermitian varieties","authors":"Angela Aguglia, Luca Giuzzi","doi":"10.1002/jcd.21870","DOIUrl":"https://doi.org/10.1002/jcd.21870","url":null,"abstract":"<p>By Aguglia et al., new quasi-Hermitian varieties <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ℳ</mi>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>,</mo>\u0000 <mi>β</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{rm{ {mathcal M} }}}_{alpha ,beta }$</annotation>\u0000 </semantics></math> in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 <mo>,</mo>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(r,{q}^{2})$</annotation>\u0000 </semantics></math> depending on a pair of parameters <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>,</mo>\u0000 <mi>β</mi>\u0000 </mrow>\u0000 <annotation> $alpha ,beta $</annotation>\u0000 </semantics></math> from the underlying field <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>GF</mtext>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>q</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{GF}({q}^{2})$</annotation>\u0000 </semantics></math> have been constructed. In the present paper we study the structure of the lines contained in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ℳ</mi>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>,</mo>\u0000 <mi>β</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{rm{ {mathcal M} }}}_{alpha ,beta }$</annotation>\u0000 </semantics></math> and consequently determine the projective equivalence classes of such varieties for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math> odd and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 <mo>=</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation> $r=3$</annotation>\u0000 </semantics></math>. As a byproduct, we also prove that the collinearity graph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 2","pages":"124-138"},"PeriodicalIF":0.7,"publicationDate":"2022-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50123664","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}
In this paper, we prove that there are no projective planes of order 12 admitting a collineation group of order 4. This yields that the order of any collineation group of a projective plane of order 12 is 1, 2, or 3.
{"title":"Projective planes of order 12 do not have a collineation group of order 4","authors":"Kenzi Akiyama, Chihiro Suetake, Masaki Tanaka","doi":"10.1002/jcd.21869","DOIUrl":"https://doi.org/10.1002/jcd.21869","url":null,"abstract":"<p>In this paper, we prove that there are no projective planes of order 12 admitting a collineation group of order 4. This yields that the order of any collineation group of a projective plane of order 12 is 1, 2, or 3.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 2","pages":"87-123"},"PeriodicalIF":0.7,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50115602","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}
<p>Square relative nonzero sum Heffter arrays, denoted by <math> <semantics> <mrow> <mi>N</mi> <msub> <mi>H</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>;</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> ${rm{N}}{{rm{H}}}_{t}(n;k)$</annotation> </semantics></math>, have been introduced as a variant of the classical concept of Heffter array. An <math> <semantics> <mrow> <mi>N</mi> <msub> <mi>H</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>;</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> ${rm{N}}{{rm{H}}}_{t}(n;k)$</annotation> </semantics></math> is an <math> <semantics> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> <annotation> $ntimes n$</annotation> </semantics></math> partially filled array with elements in <math> <semantics> <mrow> <msub> <mi>Z</mi> <mi>v</mi> </msub> </mrow> <annotation> ${{mathbb{Z}}}_{v}$</annotation> </semantics></math>, where <math> <semantics> <mrow> <mi>v</mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mi>k</mi> <mo>+</mo> <mi>t</mi> </mrow> <annotation> $v=2nk+t$</annotation> </semantics></math>, whose rows and whose columns contain <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math> filled cells, suc
平方相对非零和Heffter阵列，用N H t表示（n；k）$｛rm｛n｝｝｛rm{H｝｝｝_｛t｝（n；k）$，作为Heffter阵列的经典概念的变体而被引入。An N H t（n；k）$是n×n$ntimesn$用Z v$｛mathbb｛Z｝｝_｛v｝$中的元素部分填充的数组，其中v＝2nk+t$v＝2nk+t$，其行和列包含k$k$填充的单元格，使得每行和每列中的元素之和不同于0（模v$v$），对于不属于t阶子群的｛mathbb｛Z｝｝_｛v｝$中的每个x∈Zv$x$t$，x$x$或−x$-x$出现在数组中。本文给出了不含空单元的平方非零和Heffter阵列的直接构造，N H t（n；n）$｛rm｛n｝｝，对于每n$n$奇数，当t$t$是n$n$的除数并且当t∈{2，2n，n2，2 n 2｝$tin｛2,2n，｛n｝^｛2｝，2｛n}^｛2中｝｝$。
{"title":"Tight globally simple nonzero sum Heffter arrays and biembeddings","authors":"Lorenzo Mella, Anita Pasotti","doi":"10.1002/jcd.21866","DOIUrl":"https://doi.org/10.1002/jcd.21866","url":null,"abstract":"<p>Square relative nonzero sum Heffter arrays, denoted by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 \u0000 <msub>\u0000 <mi>H</mi>\u0000 \u0000 <mi>t</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>;</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{N}}{{rm{H}}}_{t}(n;k)$</annotation>\u0000 </semantics></math>, have been introduced as a variant of the classical concept of Heffter array. An <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 \u0000 <msub>\u0000 <mi>H</mi>\u0000 \u0000 <mi>t</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>;</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{N}}{{rm{H}}}_{t}(n;k)$</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> partially filled array with elements in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 \u0000 <mi>v</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{Z}}}_{v}$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $v=2nk+t$</annotation>\u0000 </semantics></math>, whose rows and whose columns contain <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> filled cells, suc","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 1","pages":"41-83"},"PeriodicalIF":0.7,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50151154","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}
In this paper, we give an alternative construction of the Hermitian unital 2-(28, 4, 1) design in such a way that it is constructed on the isotropic vectors in a unitary geometry of dimension 3 over the field . As a corollary, we can construct a unique 3-(10, 4, 1) design (namely, the Witt system ).
{"title":"An alternative construction of the Hermitian unital 2-(28, 4, 1) design","authors":"Koichi Inoue","doi":"10.1002/jcd.21861","DOIUrl":"https://doi.org/10.1002/jcd.21861","url":null,"abstract":"<p>In this paper, we give an alternative construction of the Hermitian unital 2-(28, 4, 1) design in such a way that it is constructed on the isotropic vectors in a unitary geometry of dimension 3 over the field <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{4}$</annotation>\u0000 </semantics></math>. As a corollary, we can construct a unique 3-(10, 4, 1) design (namely, the Witt system <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>W</mi>\u0000 \u0000 <mn>10</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{boldsymbol{W}}}_{{bf{10}}}$</annotation>\u0000 </semantics></math>).</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 12","pages":"752-759"},"PeriodicalIF":0.7,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72166349","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}
In this paper, we give an alternative construction of the Hermitian unital 2‐(28, 4, 1) design in such a way that it is constructed on the isotropic vectors in a unitary geometry of dimension 3 over the field F 4 ${{mathbb{F}}}_{4}$ . As a corollary, we can construct a unique 3‐(10, 4, 1) design (namely, the Witt system W 10 ${{boldsymbol{W}}}_{{bf{10}}}$ ).
{"title":"An alternative construction of the Hermitian unital 2‐(28, 4, 1) design","authors":"Koichi Inoue","doi":"10.1002/jcd.21861","DOIUrl":"https://doi.org/10.1002/jcd.21861","url":null,"abstract":"In this paper, we give an alternative construction of the Hermitian unital 2‐(28, 4, 1) design in such a way that it is constructed on the isotropic vectors in a unitary geometry of dimension 3 over the field F 4 ${{mathbb{F}}}_{4}$ . As a corollary, we can construct a unique 3‐(10, 4, 1) design (namely, the Witt system W 10 ${{boldsymbol{W}}}_{{bf{10}}}$ ).","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"6 1","pages":"753 - 759"},"PeriodicalIF":0.7,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86699161","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 k ‐semiframe of type g u is a k ‐GDD of type g u ( X , G , ℬ ) , in which the collection of blocks ℬ can be written as a disjoint union ℬ = P ∪ Q , where P is partitioned into parallel classes of X and Q is partitioned into holey parallel classes, each holey parallel class being a partition of X G for some G ∈ G . In this paper, we will introduce a new concept of t ‐perfect semiframe and use it to prove the existence of a 3‐semiframe of type g u with even group size. This completes the proof of the existence of 3‐semiframes with uniform group size.
g - u型k -半框架是g - u (X, g，∑)型的k - GDD，其中块的集合∑可以写成一个不相交的并集∑= P∪Q，其中P被划分为X的并行类，Q被划分为空洞并行类，每个空洞并行类是X g对某个g∈g的一个划分。在本文中，我们将引入t -完美半框架的新概念，并利用它证明了群大小为偶数的g - u型3 -半框架的存在性。这就完成了群大小一致的3 -半框存在性的证明。
{"title":"Completing the spectrum of semiframes with block size three","authors":"H. Cao, D. Xu, Hao Zheng","doi":"10.1002/jcd.21856","DOIUrl":"https://doi.org/10.1002/jcd.21856","url":null,"abstract":"A k ‐semiframe of type g u is a k ‐GDD of type g u ( X , G , ℬ ) , in which the collection of blocks ℬ can be written as a disjoint union ℬ = P ∪ Q , where P is partitioned into parallel classes of X and Q is partitioned into holey parallel classes, each holey parallel class being a partition of X G for some G ∈ G . In this paper, we will introduce a new concept of t ‐perfect semiframe and use it to prove the existence of a 3‐semiframe of type g u with even group size. This completes the proof of the existence of 3‐semiframes with uniform group size.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"5 1","pages":"716 - 732"},"PeriodicalIF":0.7,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76964572","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 -semiframe of type is a -GDD of type , in which the collection of blocks can be written as a disjoint union , where is partitioned into parallel classes of and is partitioned into holey parallel classes, each holey parallel class being a partition of for some . In this paper, we will introduce a new concept of -perfect semiframe and use it to prove the existence of a 3-semiframe of type with even group size. This completes the proof of the existence of 3-semiframes with uniform group size.
{"title":"Completing the spectrum of semiframes with block size three","authors":"H. Cao, D. Xu, H. Zheng","doi":"10.1002/jcd.21856","DOIUrl":"https://doi.org/10.1002/jcd.21856","url":null,"abstract":"<p>A <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-semiframe of type <math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>g</mi>\u0000 \u0000 <mi>u</mi>\u0000 </msup>\u0000 </mrow></math> is a <math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-GDD of type <math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>g</mi>\u0000 \u0000 <mi>u</mi>\u0000 </msup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ℬ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math>, in which the collection of blocks <math>\u0000 \u0000 <mrow>\u0000 <mi>ℬ</mi>\u0000 </mrow></math> can be written as a disjoint union <math>\u0000 \u0000 <mrow>\u0000 <mi>ℬ</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>P</mi>\u0000 \u0000 <mo>∪</mo>\u0000 \u0000 <mi>Q</mi>\u0000 </mrow></math>, where <math>\u0000 \u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow></math> is partitioned into parallel classes of <math>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow></math> and <math>\u0000 \u0000 <mrow>\u0000 <mi>Q</mi>\u0000 </mrow></math> is partitioned into holey parallel classes, each holey parallel class being a partition of <math>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo></mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow></math> for some <math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow></math>. In this paper, we will introduce a new concept of <math>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow></math>-perfect semiframe and use it to prove the existence of a 3-semiframe of type <math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>g</mi>\u0000 \u0000 <mi>u</mi>\u0000 </msup>\u0000 </mrow></math> with even group size. This completes the proof of the existence of 3-semiframes with uniform group size.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 11","pages":"716-732"},"PeriodicalIF":0.7,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72169121","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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