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}
Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite nonovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family in one of the known finite circle geometries of order , with , must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.
{"title":"Stability of Erdős–Ko–Rado theorems in circle geometries","authors":"Sam Adriaensen","doi":"10.1002/jcd.21854","DOIUrl":"https://doi.org/10.1002/jcd.21854","url":null,"abstract":"<p>Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite nonovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}$</annotation>\u0000 </semantics></math> in one of the known finite circle geometries of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>, with <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>ℱ</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mfrac>\u0000 <mn>1</mn>\u0000 \u0000 <msqrt>\u0000 <mn>2</mn>\u0000 </msqrt>\u0000 </mfrac>\u0000 \u0000 <msup>\u0000 <mi>q</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <msqrt>\u0000 <mn>2</mn>\u0000 </msqrt>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>8</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $| {rm{ {mathcal F} }},| ge frac{1}{sqrt{2}}{q}^{2}+2sqrt{2}q+8$</annotation>\u0000 </semantics></math>, must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 11","pages":"689-715"},"PeriodicalIF":0.7,"publicationDate":"2022-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72149789","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}
It is shown that, apart from the smallest Ree group, a flag‐transitive automorphism group G $G$ of a 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ design D ${mathscr{D}}$ , with λ∣k $lambda | k$ , is either an affine group or an almost simple classical group. Moreover, when G $G$ is the smallest Ree group, D ${mathscr{D}}$ is isomorphic either to the 2‐ (62,6,2) $({6}^{2},6,2)$ design or to one of the three 2‐ (62,6,6) $({6}^{2},6,6)$ designs constructed in this paper. All the four 2‐designs have the 36 secants of a non‐degenerate conic C ${mathscr{C}}$ of PG2(8) $P{G}_{2}(8)$ as a point set and 6‐sets of secants in a remarkable configuration as a block set.
{"title":"On flag‐transitive 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ designs with λ∣k $lambda | k$","authors":"Alessandro Montinaro, Eliana Francot","doi":"10.1002/jcd.21852","DOIUrl":"https://doi.org/10.1002/jcd.21852","url":null,"abstract":"It is shown that, apart from the smallest Ree group, a flag‐transitive automorphism group G $G$ of a 2‐ (k2,k,λ) $({k}^{2},k,lambda )$ design D ${mathscr{D}}$ , with λ∣k $lambda | k$ , is either an affine group or an almost simple classical group. Moreover, when G $G$ is the smallest Ree group, D ${mathscr{D}}$ is isomorphic either to the 2‐ (62,6,2) $({6}^{2},6,2)$ design or to one of the three 2‐ (62,6,6) $({6}^{2},6,6)$ designs constructed in this paper. All the four 2‐designs have the 36 secants of a non‐degenerate conic C ${mathscr{C}}$ of PG2(8) $P{G}_{2}(8)$ as a point set and 6‐sets of secants in a remarkable configuration as a block set.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"20 1","pages":"653 - 670"},"PeriodicalIF":0.7,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76703413","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>It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> of a 2-<math> <semantics> <mrow> <mo>(</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> <annotation> $({k}^{2},k,lambda )$</annotation> </semantics></math> design <math> <semantics> <mrow> <mi>D</mi> </mrow> <annotation> ${mathscr{D}}$</annotation> </semantics></math>, with <math> <semantics> <mrow> <mi>λ</mi> <mo>∣</mo> <mi>k</mi> </mrow> <annotation> $lambda | k$</annotation> </semantics></math>, is either an affine group or an almost simple classical group. Moreover, when <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> is the smallest Ree group, <math> <semantics> <mrow> <mi>D</mi> </mrow> <annotation> ${mathscr{D}}$</annotation> </semantics></math> is isomorphic either to the 2-<math> <semantics> <mrow> <mo>(</mo> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <annotation> $({6}^{2},6,2)$</annotation> </semantics></math> design or to one of the three 2-<math> <semantics> <mrow> <mo>(</mo> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <annotation> $({6}^{2},6,6)$</annotation> </semantics></math> designs constructed in this paper. All the four 2-designs have the 36 secants of a non-degenerate conic <math> <sem
<p>An idempotent Latin square of order <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math> is called resolvable and denoted by RILS(<i>v</i>) if the <math> <semantics> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $v(v-1)$</annotation> </semantics></math> off-diagonal cells can be resolved into <math> <semantics> <mrow> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> <annotation> $v-1$</annotation> </semantics></math> disjoint transversals. A large set of resolvable idempotent Latin squares of order <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math>, briefly LRILS(<i>v</i>), is a collection of <math> <semantics> <mrow> <mi>v</mi> <mo>−</mo> <mn>2</mn> </mrow> <annotation> $v-2$</annotation> </semantics></math> RILS(<i>v</i>)s pairwise agreeing on only the main diagonal. In this article, an LRILS(<i>v</i>) is constructed for <math> <semantics> <mrow> <mi>v</mi> <mo>∈</mo> <mrow> <mo>{</mo> <mrow> <mn>14</mn> <mo>,</mo> <mn>20</mn> <mo>,</mo> <mn>22</mn> <mo>,</mo> <mn>28</mn> <mo>,</mo> <mn>34</mn> <mo>,</mo> <mn>35</mn> <mo>,</mo> <mn>38</mn> <mo>,</mo> <mn>40</mn> <mo>,</mo>
{"title":"The spectrum for large sets of resolvable idempotent Latin squares","authors":"Xiangqian Li, Yanxun Chang","doi":"10.1002/jcd.21853","DOIUrl":"https://doi.org/10.1002/jcd.21853","url":null,"abstract":"<p>An idempotent Latin square of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math> is called resolvable and denoted by RILS(<i>v</i>) if the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $v(v-1)$</annotation>\u0000 </semantics></math> off-diagonal cells can be resolved into <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $v-1$</annotation>\u0000 </semantics></math> disjoint transversals. A large set of resolvable idempotent Latin squares of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>, briefly LRILS(<i>v</i>), is a collection of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $v-2$</annotation>\u0000 </semantics></math> RILS(<i>v</i>)s pairwise agreeing on only the main diagonal. In this article, an LRILS(<i>v</i>) is constructed for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mrow>\u0000 <mn>14</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>20</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>22</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>28</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>34</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>35</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>38</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>40</mn>\u0000 \u0000 <mo>,</mo>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 10","pages":"671-683"},"PeriodicalIF":0.7,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72166526","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}
An idempotent Latin square of order v $v$ is called resolvable and denoted by RILS(v) if the v(v − 1 ) $v(v-1)$ off‐diagonal cells can be resolved into v − 1 $v-1$ disjoint transversals. A large set of resolvable idempotent Latin squares of order v $v$ , briefly LRILS(v), is a collection of v − 2 $v-2$ RILS(v)s pairwise agreeing on only the main diagonal. In this article, an LRILS(v) is constructed for v ∈{14 , 20 , 22 , 28 , 34 , 35 , 38 , 40 , 42 , 46 , 50 , 55 , 62 } $vin {14,20,22,28,34,35,38,40,42,46,50,55,62}$ by using multiplier automorphism groups. Hence, there exists an LRILS(v) for any positive integer v ≥ 3 $vge 3$ , except v = 6 $v=6$ .
如果v(v-1)$ v(v-1)$离对角线单元可以分解成v-1$ v-1$不相交的截线，则v$ v$阶的幂等拉丁方阵称为可分解的，用RILS(v)表示。一个大的v$ v$阶可解幂等拉丁平方集，简称LRILS(v)，是v−2$ v-2$ RILS(v)的集合，它们只在主对角线上成对一致。本文利用乘子自同构群对v∈{14,20,22,28,34,35,38,40,42,46,50,55,62}$ v In {14,20,22,28,34,35,38,40,42,46,50,55,62}$构造了一个LRILS(v)。因此，除了v=6$ v=6$外，对于任何正整数v≥3$ v ge3 $都存在LRILS(v)。
{"title":"The spectrum for large sets of resolvable idempotent Latin squares","authors":"Xiangqian Li, Yanxun Chang","doi":"10.1002/jcd.21853","DOIUrl":"https://doi.org/10.1002/jcd.21853","url":null,"abstract":"An idempotent Latin square of order v $v$ is called resolvable and denoted by RILS(v) if the v(v − 1 ) $v(v-1)$ off‐diagonal cells can be resolved into v − 1 $v-1$ disjoint transversals. A large set of resolvable idempotent Latin squares of order v $v$ , briefly LRILS(v), is a collection of v − 2 $v-2$ RILS(v)s pairwise agreeing on only the main diagonal. In this article, an LRILS(v) is constructed for v ∈{14 , 20 , 22 , 28 , 34 , 35 , 38 , 40 , 42 , 46 , 50 , 55 , 62 } $vin {14,20,22,28,34,35,38,40,42,46,50,55,62}$ by using multiplier automorphism groups. Hence, there exists an LRILS(v) for any positive integer v ≥ 3 $vge 3$ , except v = 6 $v=6$ .","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"71 1","pages":"671 - 683"},"PeriodicalIF":0.7,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83991533","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 $k$ ‐extended q $q$ ‐near Skolem sequence of order n $n$ , denoted by Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ , is a sequence s1,s2,…,s2n−1 ${s}_{1},{s}_{2},ldots ,{s}_{2n-1}$ where sk=0 ${s}_{k}=0$ and for each integer ℓ∈[1,n]{q} $ell in [1,n]backslash {q}$ there are two indices i $i$ , j $j$ such that si=sj=ℓ ${s}_{i}={s}_{j}=ell $ and ∣i−j∣=ℓ $| i-j| =ell $ . For an Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ to exist it is necessary that q≡k(mod2) $qequiv k,(mathrm{mod},2)$ when n≡0,1(mod4) $nequiv 0,1,(mathrm{mod},4)$ and q≢k(mod2) $qnotequiv k,(mathrm{mod},2)$ when n≡2,3(mod4) $nequiv 2,3,(mathrm{mod},4)$ , where (n,q,k)≠(3,2,3) $(n,q,k)ne (3,2,3)$ , (4,2,4) $(4,2,4)$ . Any triple (n,q,k) $(n,q,k)$ satisfying these conditions is called admissible. In this manuscript, which is Part III of three manuscripts, we construct the remaining sequences; that is, Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ for all admissible (n,q,k) $(n,q,k)$ with q∈⌊n+23⌋,⌊n−22⌋ $qin left[lfloor frac{n+2}{3}rfloor ,lfloor frac{n-2}{2}rfloor right]$ and k∈⌊2n3⌋,n−1 $kin left[lfloor frac{2n}{3}rfloor ,n-1right]$ .
{"title":"Extended near Skolem sequences, Part III","authors":"C. Baker, V. Linek, N. Shalaby","doi":"10.1002/jcd.21851","DOIUrl":"https://doi.org/10.1002/jcd.21851","url":null,"abstract":"A k $k$ ‐extended q $q$ ‐near Skolem sequence of order n $n$ , denoted by Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ , is a sequence s1,s2,…,s2n−1 ${s}_{1},{s}_{2},ldots ,{s}_{2n-1}$ where sk=0 ${s}_{k}=0$ and for each integer ℓ∈[1,n]{q} $ell in [1,n]backslash {q}$ there are two indices i $i$ , j $j$ such that si=sj=ℓ ${s}_{i}={s}_{j}=ell $ and ∣i−j∣=ℓ $| i-j| =ell $ . For an Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ to exist it is necessary that q≡k(mod2) $qequiv k,(mathrm{mod},2)$ when n≡0,1(mod4) $nequiv 0,1,(mathrm{mod},4)$ and q≢k(mod2) $qnotequiv k,(mathrm{mod},2)$ when n≡2,3(mod4) $nequiv 2,3,(mathrm{mod},4)$ , where (n,q,k)≠(3,2,3) $(n,q,k)ne (3,2,3)$ , (4,2,4) $(4,2,4)$ . Any triple (n,q,k) $(n,q,k)$ satisfying these conditions is called admissible. In this manuscript, which is Part III of three manuscripts, we construct the remaining sequences; that is, Nnq(k) ${{mathscr{N}}}_{n}^{q}(k)$ for all admissible (n,q,k) $(n,q,k)$ with q∈⌊n+23⌋,⌊n−22⌋ $qin left[lfloor frac{n+2}{3}rfloor ,lfloor frac{n-2}{2}rfloor right]$ and k∈⌊2n3⌋,n−1 $kin left[lfloor frac{2n}{3}rfloor ,n-1right]$ .","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"80 1","pages":"637 - 652"},"PeriodicalIF":0.7,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81213966","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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