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

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

Alternating groups as flag-transitive automorphism groups of 2-designs with block size seven

In this article, we study flag-transitive $2$-$(v,k,\lambda)$ designs with small block size. We show that if $k$ is prime, then $G$ is point-primitive. In particular, we show that if $k=7$, then $G$ is of almost simple or affine type. We also prove that if $\mathcal{D}$ is a $2$-design with $k=7$ admitting flag-transitive almost simple automorphism group with socle an alternating group, then $\mathcal{D}$ is $PG_{2}(3,2)$ with parameter set $(15,7,3)$ and $G=A_7$, or $\mathcal{D}$ is the $2$-design with parameter set $(55, 7, 1680)$ and $G=A_{11}$ or $S_{11}$

Flag-transitive automorphism groups of 22-designs with λ≥(r,λ)2\lambda\geq (r,\lambda)^2 are not product type

In this paper we show that a flag-transitive automorphism group $G$ of a non-trivial $2$-$(v,k,\lambda)$ design with $\lambda\geq (r, \lambda)^2$ is not of product action type. In conclusion, $G$ is a primitive group of affine or almost simple type.Comment: 13 pages,2 figure

Constructing flag-transitive incidence structures

The aim of this research is to develop efficient techniques to construct flag-transitive incidence structures. In this paper we describe those techniques, present the construction results and take a closer look at how some types of flag-transitive incidence structures relate to arctransitive graphs

Designs and binary codes from maximal subgroups and conjugacy classes of ({rm M}_{11})

By using a method of construction of block-primitive and point-transitive 1-designs, in this paper we determine all block-primitive and point-transitive 1-((v, k, lambda))-designs from the maximal subgroups and the conjugacy classes of elements of the small Mathieu group ({rm M}_{11}). We examine the properties of the 1-((v, k, lambda))-designs and construct the codes defined by the binary row span of their incidence matrices. Furthermore, we present a number of interesting (Delta)-divisible binary codes invariant under ({rm M}_{11})