Journal of Combinatorial Designs最新文献

Doubly Orthogonal Equi-Squares and Sliced Orthogonal Arrays

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-04-03 DOI: 10.1002/jcd.21982

John Lorch

{"title":"Doubly Orthogonal Equi-Squares and Sliced Orthogonal Arrays","authors":"John Lorch","doi":"10.1002/jcd.21982","DOIUrl":"https://doi.org/10.1002/jcd.21982","url":null,"abstract":"<div>\u0000 \u0000 We introduce doubly orthogonal equi-squares. Using linear algebra over finite fields, we produce large families of mutually <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-doubly orthogonal equi-<math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>s</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> squares, and show these are of maximal size when <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>r</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. These results specialize to the results of Xu, Haaland, and Qian when <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>s</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and the equi-squares are Sudoku Latin squares of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Further, we show how a collection of mutually <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-doubly orthogonal equi-<math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"275-283"},"PeriodicalIF":0.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944394","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

On the Terwilliger Algebra of the Group Association Scheme of the Symmetric Group Sym ( 7 )

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-04-03 DOI: 10.1002/jcd.21981

Allen Herman, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra

{"title":"On the Terwilliger Algebra of the Group Association Scheme of the Symmetric Group \u0000 \u0000 \u0000 \u0000 Sym\u0000 \u0000 (\u0000 7\u0000 )","authors":"Allen Herman, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1002/jcd.21981","DOIUrl":"https://doi.org/10.1002/jcd.21981","url":null,"abstract":"Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, for <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>6</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, have been studied and completely determined. The case for <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>7</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is computationally much more difficult and has a potential application to find the size of the largest permutation codes of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mn>7</mn>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with a minimal distance of at least 4. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Sym</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"261-274"},"PeriodicalIF":0.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21981","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944393","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

The Directed Oberwolfach Problem With Variable Cycle Lengths: A Recursive Construction 变周期长的有向Oberwolfach问题：一个递归构造

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-03-24 DOI: 10.1002/jcd.21967

Suzan Kadri, Mateja Šajna

{"title":"The Directed Oberwolfach Problem With Variable Cycle Lengths: A Recursive Construction","authors":"Suzan Kadri, Mateja Šajna","doi":"10.1002/jcd.21967","DOIUrl":"https://doi.org/10.1002/jcd.21967","url":null,"abstract":"The directed Oberwolfach problem <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mstyle>\u0000 <mspace></mspace>\u0000 \u0000 <mtext>OP</mtext>\u0000 <mspace></mspace>\u0000 </mstyle>\u0000 \u0000 <mo>*</mo>\u0000 </msup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>m</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>m</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> asks whether the complete symmetric digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msubsup>\u0000 <mi>K</mi>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>*</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, assuming <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <msub>\u0000 <mi>m</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>m</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, admits a decomposition into spanning subdigraphs, each a disjoint un","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 7","pages":"239-260"},"PeriodicalIF":0.5,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21967","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143945026","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

Putatively Optimal Projective Spherical Designs With Little Apparent Symmetry 具有少量明显对称性的推定最优射影球面设计

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-03-11 DOI: 10.1002/jcd.21979

Alex Elzenaar, Shayne Waldron

{"title":"Putatively Optimal Projective Spherical Designs With Little Apparent Symmetry","authors":"Alex Elzenaar, Shayne Waldron","doi":"10.1002/jcd.21979","DOIUrl":"https://doi.org/10.1002/jcd.21979","url":null,"abstract":"We give some new explicit examples of putatively optimal projective spherical designs, that is, ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in general, which requires the introduction of new techniques for their construction. New examples of interest include an 11-point spherical <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-design for <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, and a 12-point spherical <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-design for <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> given by four Mercedes-Benz frames that lie on equi-isoclinic planes. The latter example shows that the set of optimal spherical designs can be uncountable. We also give results of an extensive numerical study to determine the nature of the real algebraic variety of optimal projective real spherical designs, and in particular when it is a single point (a unique design) or corresponds to an infinite family of designs.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 6","pages":"222-234"},"PeriodicalIF":0.5,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21979","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143845833","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

A Classification of the Flag-Transitive 2- ( v , 3 , λ ) Designs With v ≡ 1 , 3 ( mod 6 ) and v ≡ 6 ( mod λ ) 一类具有v≡1的2- (v, 3, λ)标志传递设计3 （mod 6）和v≡6 (modλ )

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-03-05 DOI: 10.1002/jcd.21978

Alessandro Montinaro, Eliana Francot

{"title":"A Classification of the Flag-Transitive 2-\u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 v\u0000 ,\u0000 3\u0000 ,\u0000 λ\u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 Designs With \u0000 \u0000 \u0000 \u0000 v\u0000 ≡\u0000 1\u0000 ,\u0000 3\u0000 \u0000 \u0000 (\u0000 \u0000 mod\u0000 \u0000 6\u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 and \u0000 \u0000 \u0000 \u0000 v\u0000 ≡\u0000 6\u0000 \u0000 \u0000 (\u0000 \u0000 mod\u0000 \u0000 λ\u0000 \u0000 )","authors":"Alessandro Montinaro, Eliana Francot","doi":"10.1002/jcd.21978","DOIUrl":"https://doi.org/10.1002/jcd.21978","url":null,"abstract":"<div>\u0000 \u0000 In this paper, we provide a complete classification of the 2-<math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> designs with <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>3</mn>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mn>6</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>6</mn>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> admitting a flag-transitive automorphism group non-isomorphic to a subgroup of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>A</mi>\u0000 \u0000 <mi>Γ</mi>\u0000 \u0000 <msub>\u0000 <mi>L</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 6","pages":"217-221"},"PeriodicalIF":0.5,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143846041","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

GDD Type Spanning Bipartite Block Designs GDD类型跨越二部块设计

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-02-27 DOI: 10.1002/jcd.21976

Shoko Chisaki, Ryoh Fuji-Hara, Nobuko Miyamoto

{"title":"GDD Type Spanning Bipartite Block Designs","authors":"Shoko Chisaki, Ryoh Fuji-Hara, Nobuko Miyamoto","doi":"10.1002/jcd.21976","DOIUrl":"https://doi.org/10.1002/jcd.21976","url":null,"abstract":"<div>\u0000 \u0000 There is a one-to-one correspondence between the point set of a group divisible design (GDD) with <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> groups of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> points and the edge set of a complete bipartite graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. A block of GDD corresponds to a subgraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. We show that the concurrence conditions of two points of GDD can correspond to the edge concurrence conditions of subgraphs of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 6","pages":"207-216"},"PeriodicalIF":0.5,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143845859","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

Cycle Switching in Steiner Triple Systems of Order 19 19阶Steiner三重系统的循环切换

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-02-27 DOI: 10.1002/jcd.21975

Grahame Erskine, Terry S. Griggs

{"title":"Cycle Switching in Steiner Triple Systems of Order 19","authors":"Grahame Erskine, Terry S. Griggs","doi":"10.1002/jcd.21975","DOIUrl":"https://doi.org/10.1002/jcd.21975","url":null,"abstract":"<div>\u0000 \u0000 Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> (an <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>STS</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>), yielding another <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>STS</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. This relationship may be represented by an undirected graph. An <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>STS</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> admits cycles of lengths <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>6</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>7</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. In the particular case of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 5","pages":"195-204"},"PeriodicalIF":0.5,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595458","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

More Heffter Spaces via Finite Fields 通过有限域得到更多的空间

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-02-23 DOI: 10.1002/jcd.21974

Marco Buratti, Anita Pasotti

{"title":"More Heffter Spaces via Finite Fields","authors":"Marco Buratti, Anita Pasotti","doi":"10.1002/jcd.21974","DOIUrl":"https://doi.org/10.1002/jcd.21974","url":null,"abstract":"<div>\u0000 \u0000 A <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>;</mo>\u0000 \u0000 <mi>r</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> Heffter space is a resolvable <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>v</mi>\u0000 \u0000 <mi>r</mi>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>b</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> configuration whose points form a half-set of an abelian group <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and whose blocks are all zero-sum in <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. It was recently proved that there are infinitely many orders <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> for which, given any pair <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>r</mi>\u0000 </mrow>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 5","pages":"188-194"},"PeriodicalIF":0.5,"publicationDate":"2025-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595699","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

Pairs in Nested Steiner Quadruple Systems 嵌套Steiner四重系统中的对

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-02-20 DOI: 10.1002/jcd.21973

Yeow Meng Chee, Son Hoang Dau, Tuvi Etzion, Han Mao Kiah, Wenqin Zhang

{"title":"Pairs in Nested Steiner Quadruple Systems","authors":"Yeow Meng Chee, Son Hoang Dau, Tuvi Etzion, Han Mao Kiah, Wenqin Zhang","doi":"10.1002/jcd.21973","DOIUrl":"https://doi.org/10.1002/jcd.21973","url":null,"abstract":"<div>\u0000 \u0000 Motivated by a repair problem for fractional repetition codes in distributed storage, each block of any Steiner quadruple system (SQS) of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is partitioned into two pairs. Each pair in such a partition is called a nested design pair, and its multiplicity is the number of times it is a pair in this partition. Such a partition of each block is considered a new block design called a nested SQS. Several related questions on this type of design are considered in this paper: What is the maximum multiplicity of the nested design pair with minimum multiplicity? What is the minimum multiplicity of the nested design pair with maximum multiplicity? Are there nested quadruple systems in which all the nested design pairs have the same multiplicity? Of special interest are nested quadruple systems in which all the <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfenced>\u0000 <mfrac>\u0000 <mi>v</mi>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mfenced>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> pairs are nested design pairs with the same multiplicity. Several constructions of nested quadruple systems are considered, and in particular, classic constructions of SQS are examined.\u0000 </div>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 5","pages":"177-187"},"PeriodicalIF":0.5,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595452","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

New Families of Strength-3 Covering Arrays Using Linear Feedback Shift Register Sequences 基于线性反馈移位寄存器序列的新型强度-3覆盖阵列

IF 0.5 4区数学

Journal of Combinatorial Designs Pub Date : 2025-02-14 DOI: 10.1002/jcd.21963

Kianoosh Shokri, Lucia Moura

{"title":"New Families of Strength-3 Covering Arrays Using Linear Feedback Shift Register Sequences","authors":"Kianoosh Shokri, Lucia Moura","doi":"10.1002/jcd.21963","DOIUrl":"https://doi.org/10.1002/jcd.21963","url":null,"abstract":"In an array over an alphabet of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> symbols, a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-set of column indices <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>c</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>c</mi>\u0000 \u0000 <mi>t</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is covered if each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-tuple of the alphabet occurs at least once as a row of the sub-array indexed by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>c</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>c</mi>\u0000 \u0000 <mi>t</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. A covering array, denoted by CA<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>N</mi>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 4","pages":"156-171"},"PeriodicalIF":0.5,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21963","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446750","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