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### A new biplane of order 9 with a small automorphism group

Zvonimir Janko, Tran van Trung
1986 Journal of combinatorial theory. Series A
In fact, his biplane is not self-dual and so there are exactly two biplanes with k = 13 which are known.  ...  Biplanes of order n < 8 have been classified and there are exactly 10 biplanes with those orders  . Up to now, there have been only four biplanes of order 9 known  .  ...  In fact, his biplane is not self-dual and so there are exactly two biplanes with k = 13 which are known.  ...

### Modelling biplanes on surfaces

Arthur T White
2004 European journal of combinatorics (Print)
A biplane is a geometry corresponding to a symmetric k 2 + 1, k, 2 block design. Nontrivial biplanes are known to exist only for k = 3, 4, 5, 6, 9, 11 and 13.  ...  Group difference set constructions exist for the unique biplanes having k = 3, 4, and 5; for all three biplanes having k = 6; and for one of the four biplanes having k = 9.  ...  (d) There are at least five biplanes for k = 11, and at least two for k = 13. (e) No biplanes are known to exist, for k > 13. The open cases start with k = 16, 18, 20, . . ..  ...

### Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs [article]

Akihiro Munemasa, Vladimir D. Tonchev
2020 arXiv   pre-print
The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasi-symmetric 2-(56,12,9) and 2-(57,12,11) designs with intersection  ...  There are five nonisomorphic biplanes Bi, (1 ≤ i ≤ 5) with these parameters [7, 15.8] , all five being self-dual.  ...  are exactly five biplanes with 56 points, and consequently, exactly 16 nonisomorphic 2-(45, 9, 2) designs.  ...

### On automorphism groups of a biplane (121,16,2) [article]

Dean Crnković, Doris Dumičić Danilović, Sanja Rukavina
2020 arXiv   pre-print
Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters (121,16,2).  ...  The existence of a biplane with parameters (121,16,2) is an open problem.  ...  The repetition number of each point of a symmetric design is k and every two blocks are together incident with exactly λ points.  ...

### On primitivity and reduction for flag-transitive symmetric designs

Eugenia O'Reilly Regueiro
2005 Journal of combinatorial theory. Series A
First we see what conditions are necessary for a symmetric design to admit an imprimitive, flag-transitive automorphism group.  ...  we move on to study the possibilities for a primitive, flag-transitive automorphism group, and prove that for 3, the group must be affine or almost simple, and finally we analyse the case in which a biplane  ...  Acknowledgements Much of the work in this paper was done during my Ph.D. with a grant from the Dirección General de Asuntos del Personal Académico, UNAM, under the supervision of Martin W. Liebeck.  ...

### Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle

Eugenia O'Reilly Regueiro
2005 European journal of combinatorics (Print)
In this paper we prove that there cannot be a biplane admitting a primitive, flag-transitive automorphism group of almost simple type, with alternating or sporadic socle.  ...  Liebeck, with a grant from the Dirección General de Asuntos del Personal Académico, UNAM. I am very grateful to Martin W. Liebeck for his helpful ideas and guidance.  ...  k = 11 there are five known biplanes [2, 7, 9] , and for k = 13 there are two known biplanes  , namely a biplane and its dual.  ...

### Characterizing symmetric designs by their symmetries

Eric S Lander
1988 Journal of Algebra
We settle this question, by determining all such biplanes: only five exist.  ...  A number of important symmetric (u, k, 2) designs, or biplanes, have the property that the automorphisms fixing some block B act transitively on unordered pairs of points of B.  ...  Five such biplanes are known: (1) the unique biplanes with k = 3, (2) the unique biplanes with k=4, (3) the unique biplanes with k = 5, (4) the "nicest" biplane with k = 6, namely the unique one with an  ...

### Symmetries of biplanes [article]

Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E Praeger
2020 arXiv   pre-print
In this paper, we first study biplanes D with parameters (v,k,2), where the block size k∈{13,16}. These are the smallest parameter values for which a classification is not available.  ...  In the case where k=16, we prove that |Aut(D)| divides 2^7· 3^2· 5· 7· 11· 13.  ...  The first and second authors are also grateful to Cheryl E. Praeger and Alice Devillers for supporting their visit to The University of Western Australia during July-September 2019.  ...

### Biplanes (79, 13, 2) with involutory automorphism

Ljubo Marangunić
1992 Journal of combinatorial theory. Series A
We show that each (79, 13, 2) biplane admitting an involutory automorphism is isomorphic to one of the two designs constructed by Aschbacher. !?  ...  is a 21 by 21 incidence matrix with exactly six units in each row and exactly six units in each column, Nz3 is a 21 by 15 incidence matrix with exactly five units in each row and exact1.y seven units in  ...  This is the largest set of parameters for which a biplane is known to exist. In [ 11, Aschbacher constructs two such designs with automorphism group of order 2 .5 . 11 which are dual.  ...

### Non-transversal Vectors of Some Finite Geometries [article]

Ivica Martinjak
2016 arXiv   pre-print
There is a dichotomy in the structure of biplanes of order 7 and 9 with respect to the incidence matrix symmetry.  ...  By means of associated structural invariants, we efficiently construct four biplanes of order 9 - except the one with the smallest automorphism group, that is found by Janko and Trung.  ...  The conjecture is that there are finitely many biplanes  . More about biplanes one can find in  and  while the broader context is provided in  and  .  ...

### Sesqui-arrays, a generalisation of triple arrays [article]

R. A. Bailey, Peter J. Cameron, Tomas Nilson
2018 arXiv   pre-print
We also give a construction for K×(K-1)(K-2)/2 sesqui-arrays on K(K-1)/2 letters. This construction uses biplanes.  ...  It starts with a block of a biplane and produces an array which satisfies the requirements for a sesqui-array except possibly that of having no repeated letters in a row or column.  ...  There are only finitely many biplanes known, with K = 3, 4, 5, 6, 9, 11, 13; the numbers up to isomorphism are 1, 1, 1, 3, 4, 5, 2 (there is no classification for K = 13 as yet).  ...

### Page 3483 of Mathematical Reviews Vol. , Issue 2000e [page]

2000 Mathematical Reviews
line contains exactly s + 1 points, (3) through each point there are exactly ¢ + 1 lines, and (4) for each non-incident point line pair p, L there are exactly a@ lines through p which intersect L.  ...  Summary: “The ternary codes associated with the five known biplanes of order 9 were examined using the computer language Magma.  ...

### Ovals in projective designs

E.F Assmus, J.H van Lint
1979 Journal of combinatorial theory. Series A
There are precisely five (15, 7 , 3)-designs  .  ...  One sees easily that there are 55 such ovals forming a 2-design with parameters 2 - (11, 3, 3) . In Section 6 we will make use of this result. 5. There are precisely four biplanes of order 7  .  ...

### Planes, Biplanes, and their Codes

Chester J. Salwach
1981 The American mathematical monthly
We note that, although there are infinitely many nontrivial t-designs for each t < 5, no nontrivial designs are known for t > 5.  ...  The term "balanced" refers to the property that each pair of points, or treatments, as they are called, is contained in exactly X blocks. Below are three examples of (incomplete) t-designs.  ...  The two of order 11 and two of the biplanes of order 7 are duals of each other and the rest are self-dual (they are isomorphic to their duals).  ...

### Planes, Biplanes, and their Codes

Chester J. Salwach
1981 The American mathematical monthly
We note that, although there are infinitely many nontrivial t-designs for each t < 5, no nontrivial designs are known for t > 5.  ...  The term "balanced" refers to the property that each pair of points, or treatments, as they are called, is contained in exactly X blocks. Below are three examples of (incomplete) t-designs.  ...  The two of order 11 and two of the biplanes of order 7 are duals of each other and the rest are self-dual (they are isomorphic to their duals).  ...
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