<p>In an array over an alphabet of <span></span><math> <semantics> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> </semantics></math> symbols, a <span></span><math> <semantics> <mrow> <mrow> <mi>t</mi> </mrow> </mrow> </semantics></math>-set of column indices <span></span><math> <semantics> <mrow> <mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>c</mi> <mi>t</mi> </msub> </mrow> <mo>}</mo> </mrow> </mrow> </mrow> </semantics></math> is <i>covered</i> if each <span></span><math> <semantics> <mrow> <mrow> <mi>t</mi> </mrow> </mrow> </semantics></math>-tuple of the alphabet occurs at least once as a row of the sub-array indexed by <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>…</mi> <mo>,</mo> <msub> <mi>c</mi> <mi>t</mi> </msub> </mrow> </mrow> </semantics></math>. A <i>covering array</i>, denoted by CA<span></span><math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>N</mi>
In this paper, we construct new optimal subspace designs and, consequently, new optimal codes in the sum-rank metric. We construct new 1-designs by finding sets of disjoint maximum scattered linear sets, and use these constructions to also find new -designs for . As a means of achieving this, we establish a correspondence between the metric properties of sum-rank metric codes and the geometric properties of subspace designs. Specifically, we determine the geometric counterpart of the coding-theoretic notion of generalised weights for the sum-rank metric in terms of subspace designs and determine a geometric characterisation of MSRD codes. This enables us to characterise subspace designs via their intersections with hyperplanes and via duality operations.
{"title":"On MSRD Codes, h-Designs and Disjoint Maximum Scattered Linear Sets","authors":"Paolo Santonastaso, John Sheekey","doi":"10.1002/jcd.21972","DOIUrl":"https://doi.org/10.1002/jcd.21972","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we construct new optimal subspace designs and, consequently, new optimal codes in the sum-rank metric. We construct new 1-designs by finding sets of disjoint maximum scattered linear sets, and use these constructions to also find new <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-designs for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 <mo>></mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. As a means of achieving this, we establish a correspondence between the metric properties of sum-rank metric codes and the geometric properties of subspace designs. Specifically, we determine the geometric counterpart of the coding-theoretic notion of generalised weights for the sum-rank metric in terms of subspace designs and determine a geometric characterisation of MSRD codes. This enables us to characterise subspace designs via their intersections with hyperplanes and via duality operations.</p>\u0000 </div>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 4","pages":"137-155"},"PeriodicalIF":0.5,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143447152","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}
<div> <p>It is well known that the perfect nonlinearity of a function between finite groups <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> and <span></span><math> <semantics> <mrow> <mrow> <mi>H</mi> </mrow> </mrow> </semantics></math> can be characterized by its graph in terms of relative difference set in the direct product <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>×</mo> <mi>H</mi> </mrow> </mrow> </semantics></math> (cf. [4]). Let <span></span><math> <semantics> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </semantics></math> be the infinite set of complex roots of unity. A <span></span><math> <semantics> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </semantics></math>-valued function <span></span><math> <semantics> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> </semantics></math> on an arbitrary finite group <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is associated with a finite cyclic subgroup <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>T</mi> <mi>f</mi> </msub> </mrow> </mrow> </semantics></math> in the multiplicative group of nonzero complex numbers. For a bent function <span></span><math> <semantics> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> </semantics></math> on <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> in general, its graph is not a relative difference set in the direct product <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>×</mo>
众所周知，有限群G和有限群H之间的函数的完全非线性可以用它的图用直接积G的相对差集来表示× H （cf.[4]）。设T是无穷个单位复数根的集合。任意有限群G上的一个T值函数f与一个有限循环子群相关联T f在非零复数的乘法群中。对于f在G上的弯曲函数，它的图不是直接积G × T f的相对差集。在本文中，研究了G上的弯曲函数f的图是G ×上的相对差集的充分必要条件T。旋光场及其整体基础在我们的讨论中起着重要作用。
{"title":"Bent Functions on Finite Nonabelian Groups and Relative Difference Sets","authors":"Bangteng Xu","doi":"10.1002/jcd.21970","DOIUrl":"https://doi.org/10.1002/jcd.21970","url":null,"abstract":"<div>\u0000 \u0000 <p>It is well known that the perfect nonlinearity of a function between finite groups <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> can be characterized by its graph in terms of relative difference set in the direct product <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> (cf. [4]). Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be the infinite set of complex roots of unity. A <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-valued function <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> on an arbitrary finite group <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is associated with a finite cyclic subgroup <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mi>f</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in the multiplicative group of nonzero complex numbers. For a bent function <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in general, its graph is not a relative difference set in the direct product <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 4","pages":"125-136"},"PeriodicalIF":0.5,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446694","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 article, we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.
{"title":"Extensions of Steiner Triple Systems","authors":"Giovanni Falcone, Agota Figula, Mario Galici","doi":"10.1002/jcd.21964","DOIUrl":"https://doi.org/10.1002/jcd.21964","url":null,"abstract":"<p>In this article, we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"94-108"},"PeriodicalIF":0.5,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21964","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112441","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}
In this article, we study the BM quasi-Hermitian varieties, laying in the three-dimensional Desarguesian projective space of even order. After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent, exhibiting a behavior which is strikingly different from what happens in odd characteristic. This completes the classification project started there. Here we prove more; indeed, by using previous results, we explicitly determine the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct a family of simple orthogonal arrays , with entries in , where is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed to investigate how variables in testing interact with each other.
{"title":"On Quasi-Hermitian Varieties in Even Characteristic and Related Orthogonal Arrays","authors":"Angela Aguglia, Luca Giuzzi, Alessandro Montinaro, Viola Siconolfi","doi":"10.1002/jcd.21966","DOIUrl":"https://doi.org/10.1002/jcd.21966","url":null,"abstract":"<p>In this article, we study the BM quasi-Hermitian varieties, laying in the three-dimensional Desarguesian projective space of even order. After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent, exhibiting a behavior which is strikingly different from what happens in odd characteristic. This completes the classification project started there. Here we prove more; indeed, by using previous results, we explicitly determine the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct a family of simple orthogonal arrays <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>O</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>5</mn>\u0000 </msup>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>4</mn>\u0000 </msup>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, with entries in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed to investigate how variables in testing interact with each other.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"109-122"},"PeriodicalIF":0.5,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21966","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112442","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}
Let be a prime power and be a natural number. What are the possible cardinalities of point sets in a projective plane of order , which do not intersect any line at exactly points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values for
{"title":"Avoiding Secants of Given Size in Finite Projective Planes","authors":"Tamás Héger, Zoltán Lóránt Nagy","doi":"10.1002/jcd.21968","DOIUrl":"https://doi.org/10.1002/jcd.21968","url":null,"abstract":"<div>\u0000 \u0000 <p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a prime power and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a natural number. What are the possible cardinalities of point sets <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in a projective plane of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, which do not intersect any line at exactly <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>S</mi>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>=<","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 3","pages":"83-93"},"PeriodicalIF":0.5,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143120778","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}
Peter Danziger, Melissa A. Huggan, Rehan Malik, Trent G. Marbach
<p>We consider playing the game of Tic-Tac-Toe on block designs BIBD<span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $(v,k,lambda )$</annotation> </semantics></math> and transversal designs TD<span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $(k,n)$</annotation> </semantics></math>. Players take turns choosing points and the first player to complete a block wins the game. We show that triple systems, BIBD<span></span><math> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $(v,3,lambda )$</annotation> </semantics></math>, are a first-player win if and only if <span></span><math> <semantics> <mrow> <mi>v</mi> <mo>≥</mo> <mn>5</mn> </mrow> <annotation> $vge 5$</annotation> </semantics></math>. Further, we show that for <span></span><math> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn>
There are four nonisomorphic configurations of triples that can form a triangle in a three-uniform hypergraph, where the configurations and on consist of three triples and , respectively. Denote by ex and ex the maximum number of triples in a three-uniform hypergraph on
Jingyuan Chen, Huangsheng Yu, R. Julian R. Abel, Dianhua Wu
<p>A design is said to be <i>super-simple</i> if the intersection of any two blocks has at most two elements. A design with index <span></span><math> <semantics> <mrow> <mi>λ</mi> </mrow> <annotation> $lambda $</annotation> </semantics></math> is said to be <i>completely reducible</i>, if its blocks can be partitioned into nonempty collections <span></span><math> <semantics> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>λ</mi> </mrow> <annotation> ${{mathscr{B}}}_{i},1le ile lambda $</annotation> </semantics></math>, such that each <span></span><math> <semantics> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> </mrow> <annotation> ${{mathscr{B}}}_{i}$</annotation> </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) <span></span><math> <semantics> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <annotation> $(v,4,4)$</annotation> </semantics></math> balanced incomplete block design (<span></span><math> <semantics> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <annotation> $(v,4,4)$</annotation> </semantics></math>-BIBD for short) if and only if <span></span><math> <semantics> <mrow> <mi>v</mi> <mo>≥</mo> <mn>13</mn> </mrow> <annotation> $vge 13$</annotation> </semantics></math>
( 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 $vequiv 0,1,3,4,(mathrm{mod},12),vge 12$ .
{"title":"Completely reducible super-simple \u0000 \u0000 \u0000 (\u0000 \u0000 v\u0000 ,\u0000 4\u0000 ,\u0000 4\u0000 \u0000 )\u0000 \u0000 $(v,4,4)$\u0000 -BIBDs and related constant weight codes","authors":"Jingyuan Chen, Huangsheng Yu, R. Julian R. Abel, Dianhua Wu","doi":"10.1002/jcd.21958","DOIUrl":"https://doi.org/10.1002/jcd.21958","url":null,"abstract":"<p>A design is said to be <i>super-simple</i> if the intersection of any two blocks has at most two elements. A design with index <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math> is said to be <i>completely reducible</i>, if its blocks can be partitioned into nonempty collections <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>i</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation> ${{mathscr{B}}}_{i},1le ile lambda $</annotation>\u0000 </semantics></math>, such that each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>B</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathscr{B}}}_{i}$</annotation>\u0000 </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) <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(v,4,4)$</annotation>\u0000 </semantics></math> balanced incomplete block design (<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(v,4,4)$</annotation>\u0000 </semantics></math>-BIBD for short) if and only if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>13</mn>\u0000 </mrow>\u0000 <annotation> $vge 13$</annotation>\u0000 </semantics></math>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"33 1","pages":"27-36"},"PeriodicalIF":0.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142665937","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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