62 research outputs found
Classification of flag-transitive Steiner quadruple systems
A Steiner quadruple system of order v is a 3-(v,4,1) design, and will be
denoted SQS(v). Using the classification of finite 2-transitive permutation
groups all SQS(v) with a flag-transitive automorphism group are completely
classified, thus solving the "still open and longstanding problem of
classifying all flag-transitive 3-(v,k,1) designs" for the smallest value of k.
Moreover, a generalization of a result of H. Lueneburg (1965, Math. Z. 89,
82-90) is achieved.Comment: 11 page
Almost simple groups with socle acting on Steiner quadruple systems
Let , {}, a prime power, be a projective linear
simple group. We classify all Steiner quadruple systems admitting a group
with N \leq G \leq \Aut(N). In particular, we show that cannot act as a
group of automorphisms on any Steiner quadruple system for .Comment: 5 pages; to appear in: "Journal of Combinatorial Theory, Series A
The classification of flag-transitive Steiner 3-designs
We solve the long-standing open problem of classifying all 3-(v,k,1) designs
with a flag-transitive group of automorphisms (cf. A. Delandtsheer, Geom.
Dedicata 41 (1992), p. 147; and in: "Handbook of Incidence Geometry", ed. by F.
Buekenhout, Elsevier Science, Amsterdam, 1995, p. 273; but presumably dating
back to 1965). Our result relies on the classification of the finite
2-transitive permutation groups.Comment: 27 pages; to appear in the journal "Advances in Geometry
A Census Of Highly Symmetric Combinatorial Designs
As a consequence of the classification of the finite simple groups, it has
been possible in recent years to characterize Steiner t-designs, that is
t-(v,k,1) designs, mainly for t = 2, admitting groups of automorphisms with
sufficiently strong symmetry properties. However, despite the finite simple
group classification, for Steiner t-designs with t > 2 most of these
characterizations have remained longstanding challenging problems. Especially,
the determination of all flag-transitive Steiner t-designs with 2 < t < 7 is of
particular interest and has been open for about 40 years (cf. [11, p. 147] and
[12, p. 273], but presumably dating back to 1965). The present paper continues
the author's work [20, 21, 22] of classifying all flag-transitive Steiner
3-designs and 4-designs. We give a complete classification of all
flag-transitive Steiner 5-designs and prove furthermore that there are no
non-trivial flag-transitive Steiner 6-designs. Both results rely on the
classification of the finite 3-homogeneous permutation groups. Moreover, we
survey some of the most general results on highly symmetric Steiner t-designs.Comment: 26 pages; to appear in: "Journal of Algebraic Combinatorics
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
Steiner t-designs for large t
One of the most central and long-standing open questions in combinatorial
design theory concerns the existence of Steiner t-designs for large values of
t. Although in his classical 1987 paper, L. Teirlinck has shown that
non-trivial t-designs exist for all values of t, no non-trivial Steiner
t-design with t > 5 has been constructed until now. Understandingly, the case t
= 6 has received considerable attention. There has been recent progress
concerning the existence of highly symmetric Steiner 6-designs: It is shown in
[M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial
flag-transitive Steiner 6-design can exist. In this paper, we announce that
essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008,
ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in
Computer Scienc
On the existence of block-transitive combinatorial designs
Block-transitive Steiner -designs form a central part of the study of
highly symmetric combinatorial configurations at the interface of several
disciplines, including group theory, geometry, combinatorics, coding and
information theory, and cryptography. The main result of the paper settles an
important open question: There exist no non-trivial examples with (or
larger). The proof is based on the classification of the finite 3-homogeneous
permutation groups, itself relying on the finite simple group classification.Comment: 9 pages; to appear in "Discrete Mathematics and Theoretical Computer
Science (DMTCS)
Block-Transitive Designs in Affine Spaces
This paper deals with block-transitive - designs in affine
spaces for large , with a focus on the important index case. We
prove that there are no non-trivial 5- designs admitting a
block-transitive group of automorphisms that is of affine type. Moreover, we
show that the corresponding non-existence result holds for 4- designs,
except possibly when the group is one-dimensional affine. Our approach involves
a consideration of the finite 2-homogeneous affine permutation groups.Comment: 10 pages; to appear in: "Designs, Codes and Cryptography
The Classification of Flag-transitive Steiner 4-Designs
Among the properties of homogeneity of incidence structures flag-transitivity
obviously is a particularly important and natural one. Consequently, in the
last decades also flag-transitive Steiner tdesigns (i.e. flag-transitive
t-(v,k,1) designs) have been investigated, whereas only by the use of the
classification of the finite simple groups has it been possible in recent years
to essentially characterize all flag-transitive Steiner 2-designs. However,
despite the finite simple group classification, for Steiner t-designs with
parameters t > 2 such characterizations have remained challenging open problems
for about 40 years (cf. [11, p. 147] and [12, p. 273], but presumably dating
back to around 1965). The object of the present paper is to give a complete
classification of all flag-transitive Steiner 4-designs. Our result relies on
the classification of the finite doubly transitive permutation groups and is a
continuation of the author's work [20, 21] on the classification of all
flag-transitive Steiner 3-designs.Comment: 26 pages; to appear in: "Journal of Algebraic Combinatorics
Computational complexity of reconstruction and isomorphism testing for designs and line graphs
Graphs with high symmetry or regularity are the main source for
experimentally hard instances of the notoriously difficult graph isomorphism
problem. In this paper, we study the computational complexity of isomorphism
testing for line graphs of - designs. For this class of
highly regular graphs, we obtain a worst-case running time of for bounded parameters . In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
-designs. In a second step, we show that -designs can be reconstructed
from their line graphs in polynomial-time. The first is algebraic in nature,
the second purely combinatorial. For both, profound structural knowledge in
design theory is required. Our results extend earlier complexity results about
isomorphism testing of graphs generated from Steiner triple systems and block
designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A
- …