3,469 research outputs found
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
Some new results on the self-dual [120,60,24] code
The existence of an extremal self-dual binary linear code of length 120 is a
long-standing open problem. We continue the investigation of its automorphism
group, proving that automorphisms of order 30 and 57 cannot occur. Supposing
the involutions acting fixed point freely, we show that also automorphisms of
order 8 cannot occur and the automorphism group is of order at most 120, with
further restrictions. Finally, we present some necessary conditions for the
existence of the code, based on shadow and design theory.Comment: 23 pages, 6 tables, to appear in Finite Fields and Their Application
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)
Resolvable Mendelsohn designs and finite Frobenius groups
We prove the existence and give constructions of a -fold perfect
resolvable -Mendelsohn design for any integers with such that there exists a finite Frobenius group whose kernel
has order and whose complement contains an element of order ,
where is the least prime factor of . Such a design admits as a group of automorphisms and is perfect when is a
prime. As an application we prove that for any integer in prime factorization, and any prime dividing
for , there exists a resolvable perfect -Mendelsohn design that admits a Frobenius group as a group of
automorphisms. We also prove that, if is even and divides for
, then there are at least resolvable -Mendelsohn designs that admit a Frobenius group as a group of
automorphisms, where is Euler's totient function.Comment: Final versio
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
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