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 long-standing challenging problems. Especially, the determination of all flag-transitive Steiner t-designs with 3≤t≤6 is of particular interest and has been open for about 40 years (cf. Delandtsheer (Geom. Dedicata 41, p. 147, 1992 and Handbook of Incidence Geometry, Elsevier Science, Amsterdam, 1995, p. 273), but presumably dating back to 1965). The present paper continues the author’s work (see Huber (J. Comb. Theory Ser. A 94, 180–190, 2001; Adv. Geom. 5, 195–221, 2005; J. Algebr. Comb., 2007, to appear)) 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.      

Non-trivial t-designs without repeated blocks exist for all t

We study several classes of arrays generalizing designs and orthogonal arrays. We use these to construct non-trivial t-designs without repeated blocks for all t.     

On the structure of 1-designs with at most two block intersection numbers

We introduce the notion of an unrefinable decomposition of a 1-design with at most two block intersection numbers, which is a certain decomposition of the 1-designs collection of blocks into other 1-designs. We discover an infinite family of 1-designs with at most two block intersection numbers that each have a unique unrefinable decomposition, and we give a polynomial-time algorithm to compute an unrefinable decomposition for each such design from the family. Combinatorial designs from this family include: finite projective planes of order n; SOMAs, and more generally, partial linear spaces of order (s, t) on (s + 1)² points; as well as affine designs, and more generally, strongly resolvable designs with no repeated blocks.      

Nonexistence of nontrivial tight 8-designs

Tight t-designs are t-designs whose sizes achieve the Fisher type lower bound. We give a new necessary condition for the existence of nontrivial tight designs and then use it to show that there do not exist nontrivial tight 8-designs.     

A recursive construction for simple <Emphasis Type="Italic">t</Emphasis>-designs using resolutions

This work presents a recursive construction for simple t-designs using resolutions of the ingredient designs. The result extends a construction of t-designs in our recent paper van Trung (Des Codes Cryptogr 83:493–502, 2017). Essentially, the method in van Trung (Des Codes Cryptogr 83:493–502, 2017) describes the blocks of a constructed design as a collection of block unions from a number of appropriate pairs of disjoint ingredient designs. Now, if some pairs of these ingredient t-designs have both suitable s-resolutions, then we can define a distance mapping on their resolution classes. Using this mapping enables us to have more possibilities for forming blocks from those pairs. The method makes it possible for constructing many new simple t-designs. We give some application results of the new construction.     

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 flag-transitive Steinert-designs (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. 2000 Mathematics Subject Classification. Primary 51E10 . Secondary 05B05 . 20B25     

On lower bounds on the size of designs in compact symmetric spaces of rank 1

The concept of t-designs in compact symmetric spaces of rank 1 is a generalization of the theory of classical t-designs. In this paper we obtain new lower bounds on the cardinality of designs in projective compact symmetric spaces of rank 1. With one exception our bounds are the first improvements of the classical bounds by more than one. We use the linear programming technique and follow the approach we have proposed for spherical codes and designs. Some examples are shown and compared with the classical bounds.     

Infinite families of 3-designs from a type of five-weight code

It has been known for a long time that t-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a t-design. While a lot of progress in the direction of constructing codes from t-designs has been made, only a small amount of work on the construction of t-designs from codes has been done. The objective of this paper is to construct infinite families of 2-designs and 3-designs from a type of binary linear codes with five weights. The total number of 2-designs and 3-designs obtained in this paper are exponential in any odd m and the block size of the designs varies in a huge range.     

Unifying some known infinite families of combinatorial 3-designs

In this paper we present a construction of 3-designs by using a 3-design with resolvability. The basic construction generalizes a well-known construction of simple 3-(v,4,3) designs by Jungnickel and Vanstone (1986). We investigate the conditions under which the designs obtained by the basic construction are simple. Many infinite families of simple 3-designs are presented, which are closely related to some known families by Iwasaki and Meixner (1995), Laue (2004) and van Tran (2000, 2001). On the other hand, the designs obtained by the basic construction possess various properties: A theory of constructing simple cyclic 3-(v,4,3) designs by Köhler (1981) can be readily rebuilt from the context of this paper. Moreover many infinite families of simple resolvable 3-designs are presented in comparison with some known families. We also show that for any prime power q and any odd integer n there exists a resolvable 3-(qn+1,q+1,1) design. As far as the authors know, this is the first and the only known infinite family of resolvable t-(v,k,1) designs with t?3 and k?5. Those resolvable designs can again be used to obtain more infinite families of simple 3-designs through the basic construction.     

Relative <Emphasis Type="Italic">t</Emphasis>-designs in binary Hamming association scheme <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, 2)

A relative t-design in the binary Hamming association schemes H(n, 2) is equivalent to a weighted regular t-wise balanced design, i.e., certain combinatorial t-design which allows different sizes of blocks and a weight function on blocks. In this paper, we study relative t-designs in H(n, 2), putting emphasis on Fisher type inequalities and the existence of tight relative t-designs. We mostly consider relative t-designs on two shells. We prove that if the weight function is constant on each shell of a relative t-design on two shells then the subset in each shell must be a combinatorial \((t-1)\)-design. This is a generalization of the result of Kageyama who proved this under the stronger assumption that the weight function is constant on the whole block set. Using this, we define tight relative t-designs for odd t, and a strong restriction on the possible parameters of tight relative t-designs in H(n, 2). We obtain a new family of such tight relative t-designs, which were unnoticed before. We will give a list of feasible parameters of such relative 3-designs with \(n \le 100\), and then we discuss the existence and/or the non-existence of such tight relative 3-designs. We also discuss feasible parameters of tight relative 4-designs on two shells in H(n, 2) with \(n \le 50\). In this study we come up with the connection on the topics of classical design theory, such as symmetric 2-designs (in particular 2-\((4u-1,2u-1,u-1)\) Hadamard designs) and Driessen’s result on the non-existence of certain 3-designs. We believe Problems 1 and 2 presented in Sect. 5.2 open a new way to study relative t-designs in H(n, 2). We conclude our paper listing several open problems.     

More on block intersection polynomials and new applications to graphs and block designs

The concept of intersection numbers of order r for t-designs is generalized to graphs and to block designs which are not necessarily t-designs. These intersection numbers satisfy certain integer linear equations involving binomial coefficients, and information on the non-negative integer solutions to these equations can be obtained using the block intersection polynomials introduced by P.J. Cameron and the present author. The theory of block intersection polynomials is extended, and new applications of these polynomials to the studies of graphs and block designs are obtained. In particular, we obtain a new method of bounding the size of a clique in an edge-regular graph with given parameters, which can improve on the Hoffman bound when applicable, and a new method for studying the possibility of a graph with given vertex-degree sequence being an induced subgraph of a strongly regular graph with given parameters.     

On the Cameron-Praeger conjecture

This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron-Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6-(v,k,λ) designs with λ=1, except possibly when the group is PΓL(2,pe) with p=2 or 3, and e is an odd prime power.     

A few more large sets of t-designs

We construct several new large sets of t-designs that are invariant under Frobenius groups, and discuss their consequences. These large sets give rise to further new large sets by means of known recursive constructions including an infinite family of large sets of 3 − (v, 4, λ) designs. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 293–308, 1998     

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 t-(v,k,λ) designs. For this class of highly regular graphs, we obtain a worst-case running time of O(vlogv+O(1)) for bounded parameters t, k, λ.In a first step, our approach makes use of the Babai-Luks algorithm to compute canonical forms of t-designs. In a second step, we show that t-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.     

Polynomial techniques for investigation of spherical designs

We investigate the structure of spherical τ-designs by applying polynomial techniques for investigation of some inner products of such designs. Our approach can be used for large variety of parameters (dimension, cardinality, strength). We obtain new upper bounds for the largest inner product, lower bounds for the smallest inner product and some other bounds. Applications are shown for proving nonexistence results either in small dimensions and in certain asymptotic process. In particular, we complete the classification of the cardinalities for which 3-designs on exist for n = 8, 13, 14 and 18. We also obtain new asymptotic lower bound on the minimum possible odd cardinality of 3-designs.      

Extending t-designs

Three extension theorems for t-designs are proved; two for t even, and one for t odd. Another theorem guaranteeing that certain t-designs be (t + 1)-designs is presented. The extension theorem for odd t is used to show that every group of odd order 2k + 1, k ≠ 2^r ? 1, acts as an automorphism group of a 2-(2k + 2, k + 1, λ) design consisting of exactly one half of the (k + 1)-settled, Although the question of the existence of a 6-(14, 7, 4) design is not settled, certain requisite properties of the 4-designs on 12 elements derived from such a design are established. All of these results depend heavily upon generalizations of block intersection number equations of N. S. Mendelsohn.     

Some Indecomposable t-Designs

The existence of large sets of 5-(14,6,3) designs is in doubt. There are five simple 5-(14,6,6) designs known in the literature. In this note, by the use of a computer program, we show that all of these designs are indecomposable and therefore they do not lead to large sets of 5-(14,6,3) designs. Moreover, they provide the first counterexamples for a conjecture on disjoint t-designs which states that if there exists a t-(v, k, λ) design (X, D) with minimum possible value of λ, then there must be a t-(v, k, λ) design (X, D′) such that D∩ D′ = Ø.     

Incidence matrices of t-designs

We point out a generalization of the matrix equation NN^T=(r? λ)I+λJ to t-designs with t>2 and derive extensions of Fisher's, Connor's, and Mann's inequalities for block designs.     

Studying designs via multisets

A new method to study families of finite sets, in particular t-designs, by studying families of multisets (also called lists) and their relationships with families of sets, is developed. Notion of the tag for a subset defined earlier by one of the authors is extended to a submultiset. A new concept t-(v, k, λ) list design is defined and studied. Basic existence theory for designs is extended to a new set up of list designs. In particular tags are used to prove that signed t-list designs exist whenever necessary conditions are satisfied. The concepts of homomorphisms and block spreading are extended to this new set up.     

Systematic Construction of <Emphasis Type="Italic">q</Emphasis>-Analogs of <Emphasis Type="Italic">t</Emphasis>−(<Emphasis Type="Italic">v</Emphasis>,<Emphasis Type="Italic">k</Emphasis>,λ)-Designs

In the present paper we consider a q-analog of t–(v,k,)-designs. It is canonic since it arises by replacing sets by vector spaces over GF(q), and their orders by dimensions. These generalizations were introduced by Thomas [Geom.Dedicata vol. 63, pp. 247–253 (1996)] they are called t –(v,k,;q)- designs. A few of such q-analogs are known today, they were constructed using sophisticated geometric arguments and case-by-case methods. It is our aim now to present a general method that allows systematically to construct such designs, and to give complete catalogs (for small parameters, of course) using an implemented software package. &nbsp; In order to attack the (highly complex) construction, we prepare them for an enormous data reduction by embedding their definition into the theory of group actions on posets, so that we can derive and use a generalization of the Kramer-Mesner matrix for their definition, together with an improved version of the LLL-algorithm. By doing so we generalize the methods developed in a research project on t –(v,k,)-designs on sets, obtaining this way new results on the existence of t–(v,k,;q)-designs on spaces for further quintuples (t,v,k,;q) of parameters. We present several 2–(6,3,;2)-designs, 2–(7,3,;2)-designs and, as far as we know, the very first 3-designs over GF(q).classification 05B05