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A new biplane of order 9 with a small automorphism group

1986
*
Journal of combinatorial theory. Series A
*

In fact, his

doi:10.1016/0097-3165(86)90103-2
fatcat:7625gwkw75b3jasxwrzc2snbnq
*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 [2] . Up to now,*there*have been only four*biplanes*of order 9 known [3] . ... 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

2004
*
European journal of combinatorics (Print)
*

A

doi:10.1016/j.ejc.2003.05.004
fatcat:troioelcsrfj5dxwho37out7b4
*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]

2020
*
arXiv
*
pre-print

The dual codes of the ternary linear codes of the residual designs of

arXiv:2003.04453v3
fatcat:h3tmspzqbnarxlh5bgdrqmq6uy
*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]

2020
*
arXiv
*
pre-print

Further, we study a possible action of an automorphism of order

arXiv:2010.12944v1
fatcat:jmlhiuh4pjazxg4jt45f4k4avq
*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

2005
*
Journal of combinatorial theory. Series A
*

First we see what conditions

doi:10.1016/j.jcta.2004.08.002
fatcat:mvch2ybgurfdnap47o7tpoedhu
*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

2005
*
European journal of combinatorics (Print)
*

In this paper we prove that

doi:10.1016/j.ejc.2004.05.003
fatcat:7hwijx77kzfv7cuvjsnlksrwse
*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*[1] , namely a*biplane*and its dual. ...##
###
Characterizing symmetric designs by their symmetries

1988
*
Journal of Algebra
*

We settle this question, by determining all such

doi:10.1016/0021-8693(88)90180-9
fatcat:rmkw2smovfgblkpt7szjfcjdja
*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]

2020
*
arXiv
*
pre-print

In this paper, we first study

arXiv:2004.04535v1
fatcat:bvrdg4r32faztl5nffvxg4qg7e
*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

1992
*
Journal of combinatorial theory. Series A
*

We show that each (79, 13, 2)

doi:10.1016/0097-3165(92)90051-u
fatcat:qae5t5a5dbc3pbecnxuml62rxe
*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]

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*[3] . More about

*biplanes*one can find in [7] and [14] while the broader context is provided in [16] and [

*11*] . ...

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

2018
*
arXiv
*
pre-print

We also give a construction for

arXiv:1706.02930v2
fatcat:psauqepskjhr3mknhq2mw2vrbu
*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

1979
*
Journal of combinatorial theory. Series A
*

*There*

*are*precisely

*five*(15, 7 , 3)-designs [5] . ... 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 [12] . ...

##
###
Planes, Biplanes, and their Codes

1981
*
The American mathematical monthly
*

We note that, although

doi:10.1080/00029890.1981.11995202
fatcat:b3ojz4ha65hvlpgwiexuoupqvu
*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

1981
*
The American mathematical monthly
*

We note that, although

doi:10.2307/2321134
fatcat:uri47qlrpzhatn5ezkivljhdoq
*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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