An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance and length 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 -ary quasigroups.
{"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}
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 $tlt 8$ .
一个阿贝尔群G $G$的子集A $A$是可序的，如果它的元素有一个序(A 1，…，A k) $({a}_{1},ldots ,{a}_{k})$，使得部分和(s 0, s 1，…，s k) $({s}_{0},{s}_{1},ldots ,{s}_{k})$ (s 0 = 0 ${s}_{0}=0$和si =∑j = 1 i A j ${s}_{i}={sum }_{j=1}^{i}{a}_{j}$对于1≤i≤k $1le ile k$)是不同的，除了可能的例外，我们可以有s k = s 0 = 0 ${s}_{k}={s}_{0}=0$。在文献中，关于阿贝尔群子集的可序列性有几个猜想和问题，Alspach和Liversidge将这些猜想组合并总结为“如果一个阿贝尔群的子集不包含0，那么它是可序列的”的猜想。如果一个可序列集合a $A$的元素和不等于0，那么在Cayley图Cay [G:±a] $Cay[G:pm A]$中存在一条简单路径P $P$，使得Δ (P) =±a ${rm{Delta }}(P)=pm A$。在本文中，受这个图论解释的启发，我们提出了一个弱化这个猜想的方法。这里，在上述假设下，我们想要找到一个排序，它的部分和定义了一个周长大于t $t$的行走W $W$(对于给定的t < k $tlt k$)，并且使得Δ (W) =±a ${rm{Delta }}(W)=pm A$。这是可能的，当i $i$和j $j$不同且∣i−j∣≤t $| i-j| le t$时，部分和si ${s}_{i}$和s j ${s}_{j}$是不同的。在这种情况下，我们说集合A $A$是t $t$‐弱可测序的。这里给出的主要结果是，当t < 7 $tlt 7$或当A {}$A$不包含类型x， - x${x,-x}$和t < 8 $tlt 8$的对时，Z p⧹0${{mathbb{Z}}}_{p}setminus {0}$的任何子集A {}$A$都是t $t$‐弱可测序的。
{"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}
<p>In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2-chromatic Steiner quadruple system of order <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math> (SQS<math> <semantics> <mrow> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <annotation> $(v)$</annotation> </semantics></math>) for all <math> <semantics> <mrow> <mi>v</mi> <mo>≡</mo> <mn>4</mn> </mrow> <annotation> $vequiv 4$</annotation> </semantics></math> or <math> <semantics> <mrow> <mn>8</mn> <mspace></mspace> <mrow> <mo>(</mo> <mrow> <mi>mod</mi> <mspace></mspace> <mn>12</mn> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $8,(mathrm{mod},12)$</annotation> </semantics></math>. The first author presented a construction for 2-chromatic SQSs based on 2-chromatic candelabra quadruple systems and <math> <semantics> <mrow> <mi>s</mi> </mrow> <annotation> $s$</annotation> </semantics></math>-fan designs. In this paper, it is proved that a 2-chromatic SQS<math> <semantics> <mrow> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <annotation> $(v)$</annotation> </semantics></math> also exists if <math> <semantics> <mrow> <mrow> <mi>v</mi> <mo>≡</mo> <mn>10</mn> <mspace></mspace> <mrow> <mo>(</mo> <mrow> <mi>mod</mi> <mspace></mspace> <mn>12</mn> </mrow>
{"title":"New infinite classes of 2-chromatic Steiner quadruple systems","authors":"Lijun Ji, Shuangqing Liu, Ye Yang","doi":"10.1002/jcd.21845","DOIUrl":"https://doi.org/10.1002/jcd.21845","url":null,"abstract":"<p>In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2-chromatic Steiner quadruple system of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math> (SQS<math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v)$</annotation>\u0000 </semantics></math>) for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation> $vequiv 4$</annotation>\u0000 </semantics></math> or <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>8</mn>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mn>12</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $8,(mathrm{mod},12)$</annotation>\u0000 </semantics></math>. The first author presented a construction for 2-chromatic SQSs based on 2-chromatic candelabra quadruple systems and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mi>s</mi>\u0000 </mrow>\u0000 <annotation> $s$</annotation>\u0000 </semantics></math>-fan designs. In this paper, it is proved that a 2-chromatic SQS<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v)$</annotation>\u0000 </semantics></math> also exists if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>≡</mo>\u0000 \u0000 <mn>10</mn>\u0000 <mspace></mspace>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 \u0000 <mn>12</mn>\u0000 </mrow>\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}
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 ) $vequiv 2,(mathrm{mod},24)$ .
{"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}
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.
{"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}
A Kirkman triple system of order , KTS, is a resolvable Steiner triple system on elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS which contain as a subdesign a Steiner triple system of order , an STS. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case , 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 to (at present) three possible exceptions. In addition, we obtain results for other cases of the form and also near . 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.
{"title":"An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns","authors":"Peter J. Dukes, 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}
Steiner triple systems form one of the most studied classes of combinatorial designs. Configurations, including subsystems, play a central role in the investigation of Steiner triple systems. With sporadic instances of small systems, ad hoc algorithms for counting or listing configurations are typically fast enough for practical needs, but with many systems or large systems, the relevance of computational complexity and algorithms of low complexity is highlighted. General theoretical results as well as specific practical algorithms for important configurations are presented.
{"title":"Algorithms and complexity for counting configurations in Steiner triple systems","authors":"Daniel Heinlein, Patric R. J. Östergård","doi":"10.1002/jcd.21839","DOIUrl":"https://doi.org/10.1002/jcd.21839","url":null,"abstract":"<p>Steiner triple systems form one of the most studied classes of combinatorial designs. Configurations, including subsystems, play a central role in the investigation of Steiner triple systems. With sporadic instances of small systems, ad hoc algorithms for counting or listing configurations are typically fast enough for practical needs, but with many systems or large systems, the relevance of computational complexity and algorithms of low complexity is highlighted. General theoretical results as well as specific practical algorithms for important configurations are presented.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 7","pages":"527-546"},"PeriodicalIF":0.7,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21839","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72156027","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}
A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order ≤ 8 was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.
{"title":"Classification of minimal blocking sets in small Desarguesian projective planes","authors":"K. Coolsaet, Arne Botteldoorn, V. Fack","doi":"10.1002/jcd.21842","DOIUrl":"https://doi.org/10.1002/jcd.21842","url":null,"abstract":"A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order ≤ 8 was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"106 1","pages":"561 - 580"},"PeriodicalIF":0.7,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72870562","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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