The four known biplanes with <i>k</i> = 11

Salwach, Chester J.; Mezzaroba, Joseph A.

doi:10.1155/s0161171279000235

Cited by 14 publications

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“…It is known that there are exactly four symmetric (91,10,1) designs (see [14], [22]). Furthermore, there are exactly four symmetric (56,11,2) designs having an automorphism group of order six (see [12], [19]). Actually, all these designs have an automorphism group isomorphic to Z 6 .…”

Section: Introductionmentioning

confidence: 99%

A classification of all symmetric block designs of order nine with an automorphism of order six

Crnković

Rukavina

Schmidt

2005

J of Combinatorial Designs

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Abstract:We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. # 2005 Wiley Periodicals, Inc.

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Section: Introductionmentioning

confidence: 99%

A classification of all symmetric block designs of order nine with an automorphism of order six

Crnković

Rukavina

Schmidt

2005

J of Combinatorial Designs

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“…There are five nonisomorphic biplanes Bi, (1 ≤ i ≤ 5) with these parameters [6, 15.8], all five being self-dual. The first biplane, B1, was found by Hall, Lane and Wales [8], B2 was found by Mezzaroba and Salwach [16], B3 and B4 were found by Denniston [5], and B5 was found by Janko and Trung [11]. The residual 2-(45, 9, 2) designs of the five biplanes fall into 16 isomorphism classes (see [13,Table 2]).…”

Section: Introductionmentioning

confidence: 99%

Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs

Munemasa¹,

Tonchev²

2020

Preprint

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There are exactly five biplanes with k = 11

Kaski

Östergård²

2007

J of Combinatorial Designs

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A biplane is a 2-(k(k − 1)/2 + 1, k, 2) symmetric design. Only sixteen nontrivial biplanes are known: there are exactly nine biplanes with k < 11, at least five biplanes with k = 11, and at least two biplanes with k = 13. It is here shown by exhaustive computer search that the list of five known biplanes with k = 11 is complete. This result further implies that there exists no 3-(57, 12, 2) design, no 112 11 symmetric configuration, and no (324, 57, 0, 12) strongly regular graph. The five biplanes have 16 residual designs, which by the Hall-Connor theorem constitute a complete classification of the 2-(45, 9, 2) designs.

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The four known biplanes with k = 11

Cited by 14 publications

References 4 publications

A classification of all symmetric block designs of order nine with an automorphism of order six

A classification of all symmetric block designs of order nine with an automorphism of order six

Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs

There are exactly five biplanes with k = 11

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