Journal of Combinatorial Designs最新文献

{"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}

{"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}

{"title":"Extended near Skolem sequences, Part III","authors":"Catharine A. Baker, Vaclav Linek, Nabil Shalaby","doi":"10.1002/jcd.21851","DOIUrl":"https://doi.org/10.1002/jcd.21851","url":null,"abstract":"<p>A <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-extended <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>-near Skolem sequence of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>, denoted by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>N</mi>\u0000 <mi>n</mi>\u0000 <mi>q</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${{mathscr{N}}}_{n}^{q}(k)$</annotation>\u0000 </semantics></math>, is a sequence <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${s}_{1},{s}_{2},ldots ,{s}_{2n-1}$</annotation>\u0000 </semantics></math> where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation> ${s}_{k}=0$</annotation>\u0000 </semantics></math> and for each integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mo></mo>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mi>q</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $ell in [1,n]backslash {q}$</annotation>\u0000 </semantics></math> there are two indices <math>\u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 10","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":"72165250","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}

{"title":"Embedding in MDS codes and Latin cubes","authors":"Vladimir N. Potapov","doi":"10.1002/jcd.21849","DOIUrl":"https://doi.org/10.1002/jcd.21849","url":null,"abstract":"<p>An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance <math>\u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math> and length <math>\u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> can be embedded into an maximum distance separable (MDS) code with the same code distance and length but under a larger alphabet. As a corollary we obtain embeddings of systems of partial mutually orthogonal Latin cubes and <math>\u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-ary quasigroups.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 9","pages":"626-633"},"PeriodicalIF":0.7,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72159165","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}

{"title":"Weak sequenceability in cyclic groups","authors":"Simone Costa, Stefano Della Fiore","doi":"10.1002/jcd.21862","DOIUrl":"https://doi.org/10.1002/jcd.21862","url":null,"abstract":"A subset A $A$ of an abelian group G $G$ is sequenceable if there is an ordering ( a 1 , … , a k ) $({a}_{1},ldots ,{a}_{k})$ of its elements such that the partial sums ( s 0 , s 1 , … , s k ) $({s}_{0},{s}_{1},ldots ,{s}_{k})$ , given by s 0 = 0 ${s}_{0}=0$ and s i = ∑ j = 1 i a j ${s}_{i}={sum }_{j=1}^{i}{a}_{j}$ for 1 ≤ i ≤ k $1le ile k$ , are distinct, with the possible exception that we may have s k = s 0 = 0 ${s}_{k}={s}_{0}=0$ . In the literature there are several conjectures and questions concerning the sequenceability of subsets of abelian groups, which have been combined and summarized by Alspach and Liversidge into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable. If the elements of a sequenceable set A $A$ do not sum to 0 then there exists a simple path P $P$ in the Cayley graph C a y [ G : ± A ] $Cay[G:pm A]$ such that Δ ( P ) = ± A ${rm{Delta }}(P)=pm A$ . In this paper, inspired by this graph–theoretical interpretation, we propose a weakening of this conjecture. Here, under the above assumptions, we want to find an ordering whose partial sums define a walk W $W$ of girth bigger than t $t$ (for a given t < k $tlt k$ ) and such that Δ ( W ) = ± A ${rm{Delta }}(W)=pm A$ . This is possible given that the partial sums s i ${s}_{i}$ and s j ${s}_{j}$ are different whenever i $i$ and j $j$ are distinct and ∣ i − j ∣ ≤ t $| i-j| le t$ . In this case, we say that the set A $A$ is t $t$ ‐weakly sequenceable. The main result here presented is that any subset A $A$ of Z p ⧹ { 0 } ${{mathbb{Z}}}_{p}setminus {0}$ is t $t$ ‐weakly sequenceable whenever t < 7 $tlt 7$ or when A $A$ does not contain pairs of type { x , − x } ${x,-x}$ and t < 8 $tlt 8$ .","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"4 1","pages":"735 - 751"},"PeriodicalIF":0.7,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87428608","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}

{"title":"New infinite classes of 2-chromatic Steiner quadruple systems","authors":"Lijun Ji,&nbsp;Shuangqing Liu,&nbsp;Ye Yang","doi":"10.1002/jcd.21845","DOIUrl":"https://doi.org/10.1002/jcd.21845","url":null,"abstract":"&lt;p&gt;In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2-chromatic Steiner quadruple system of order &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (SQS&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $(v)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) for all &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≡&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $vequiv 4$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; or &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;8&lt;/mn&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;mod&lt;/mi&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 &lt;mn&gt;12&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $8,(mathrm{mod},12)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The first author presented a construction for 2-chromatic SQSs based on 2-chromatic candelabra quadruple systems and &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-fan designs. In this paper, it is proved that a 2-chromatic SQS&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $(v)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; also exists if &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≡&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;10&lt;/mn&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;mod&lt;/mi&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 \u0000 &lt;mn&gt;12&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 9","pages":"613-620"},"PeriodicalIF":0.7,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72190472","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}

{"title":"New infinite classes of 2‐chromatic Steiner quadruple systems","authors":"L. Ji, Shuangqing Liu, Ye Yang","doi":"10.1002/jcd.21845","DOIUrl":"https://doi.org/10.1002/jcd.21845","url":null,"abstract":"In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2‐chromatic Steiner quadruple system of order v $v$ (SQS ( v ) $(v)$ ) for all v ≡ 4 $vequiv 4$ or 8 ( mod 12 ) $8,(mathrm{mod},12)$ . The first author presented a construction for 2‐chromatic SQSs based on 2‐chromatic candelabra quadruple systems and s $s$ ‐fan designs. In this paper, it is proved that a 2‐chromatic SQS ( v ) $(v)$ also exists if v ≡ 10 ( mod 12 ) $vequiv 10,(mathrm{mod},12)$ , or if v ≡ 2 ( mod 24 ) $vequiv 2,(mathrm{mod},24)$ .","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"9 1","pages":"613 - 620"},"PeriodicalIF":0.7,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84180073","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}

{"title":"An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns","authors":"P. Dukes, E. Lamken","doi":"10.1002/jcd.21844","DOIUrl":"https://doi.org/10.1002/jcd.21844","url":null,"abstract":"A Kirkman triple system of order v , KTS (v) , is a resolvable Steiner triple system on v elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS (v) which contain as a subdesign a Steiner triple system of order u , an STS (u) . We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case v= 2u+ 1 , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case v= 2u+ 3 to (at present) three possible exceptions. In addition, we obtain results for other cases of the form v= 2u+w and also near v= 3u . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 1","pages":"581 - 608"},"PeriodicalIF":0.7,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88790045","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}

{"title":"The existence of partitioned balanced tournament designs","authors":"Makoto Araya,&nbsp;Naoya Tokihisa","doi":"10.1002/jcd.21846","DOIUrl":"https://doi.org/10.1002/jcd.21846","url":null,"abstract":"&lt;p&gt;E. R. Lamken proved that there exists a partitioned balanced tournament design of side &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, PBTD(&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;), for &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; a positive integer, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $nge 5$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, except possibly for &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;9&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;11&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;15&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $nin {9,11,15}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this article, we establish the existence of PBTD(&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) for &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;9&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;11&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;15&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $nin {9,11,15}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. As a consequence, the existence of PBTD(&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 9","pages":"621-625"},"PeriodicalIF":0.7,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72154677","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}

{"title":"An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns","authors":"Peter J. Dukes,&nbsp;Esther R. Lamken","doi":"10.1002/jcd.21844","DOIUrl":"https://doi.org/10.1002/jcd.21844","url":null,"abstract":"<p>A Kirkman triple system of order <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow></math>, KTS<math>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow></math>, is a resolvable Steiner triple system on <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow></math> elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS<math>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow></math> which contain as a subdesign a Steiner triple system of order <math>\u0000 <mrow>\u0000 <mi>u</mi>\u0000 </mrow></math>, an STS<math>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>u</mi>\u0000 <mo>)</mo>\u0000 </mrow></math>. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 <mi>u</mi>\u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>, for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 <mi>u</mi>\u0000 <mo>+</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow></math> to (at present) three possible exceptions. In addition, we obtain results for other cases of the form <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 \u0000 <mn>2</mn>\u0000 <mi>u</mi>\u0000 <mo>+</mo>\u0000 <mi>w</mi>\u0000 </mrow></math> and also near <math>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>=</mo>\u0000 \u0000 <mn>3</mn>\u0000 <mi>u</mi>\u0000 </mrow></math>. Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 8","pages":"581-608"},"PeriodicalIF":0.7,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72179676","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}