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

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 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

Classification of Cyclic Steiner Quadruple Systems

The problem of classifying cyclic Steiner quadruple systems (CSQSs) is considered. A computational approach shows that the number of isomorphism classes of such designs with orders 26 and 28 is 52,170 and 1,028,387, respectively. It is further shown that CSQSs of order 2p, where p is a prime, are isomorphic iff they are multiplier equivalent. Moreover, no CSQSs of order less than or equal to 38 are isomorphic but not multiplier equivalent

On the resolutions of cyclic Steiner triple systems with small parameters

The paper presents useful invariants of resolutions of cyclic $STS(v)$ with $v\le 39$, namely of all resolutions of cyclic $STS(15)$, $STS(21)$ and $STS(27)$, of the resolutions with nontrivial automorphisms of cyclic $STS(33)$ and of resolutions with automorphisms of order $13$ of cyclic $STS(39)$

On the primarity of some block intersection graphs

Philosophiae Doctor - PhDA tactical con guration consists of a nite set V of points, a nite set B of blocks and an incidence relation between them, so that all blocks are incident with the same number k points, and all points are incident with the same number r of blocks (See [14] for example ). If v := jV j and b := jBj; then v; k; b; r are known as the parameters of the con guration. Counting incident point-block pairs, one sees that vr = bk: In this thesis, we generalize tactical con gurations on Steiner triple systems obtained from projective geometry. Our objects are subgeometries as blocks. These subgeometries are collected into systems and we study them as designs and graphs. Considered recursively is a further tactical con guration on some of the designs obtained and in what follows, we obtain similar structures as the Steiner triple systems from projective geometry. We also study these subgeometries as factorizations and examine the automorphism group of the new structures. These tactical con gurations at rst sight do not form interesting structures. However, as will be shown, they o er some level of intriguing symmetries. It will be shown that they inherit the automorphism group of the parent geometry