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Representing a design graphically can make certain properties of a design obvious that otherwise would require tedious reasoning. This holds for symmetries, resolvability, and subdesign inclusion in appropriate cases. We describe how a graphical representation can be achieved in many cases and show several examples, in particular visualizations of the biplanes on up to 16 vertices and subdesigns of the large Witt design. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:1–22, 2012

In the present paper we consider a q-analog of t−(v,k,λ)-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,λ;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.   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,λ)-designs on sets, obtaining this way new results on the existence of t−(v,k,λ;q)-designs on spaces for further quintuples (t,v,k,λ;q) of parameters. We present several 2−(6,3,λ;2)-designs, 2−(7,3,λ;2)-designs and, as far as we know, the very first 3-designs over GF(q).

A general group theoretic approach is used to find resolvable designs. Infinitely many resolvable 3-designs are obtained where each is block transitive under some PSL(2, pf) or PGL(2, pf). Some known Steiner 5-designs are assembled from such resolvable 3-designs such that they are also resolvable. We give some visualizations of Steiner systems which make resolvability obvious.

The MOLGEN Chemical Identifier MOLGEN-CID is a software module freely accessible via the Internet. For a molecule or graph entered in molfile format (2D) it produces, by a canonical renumbering procedure, a canonical molfile and a unique character string that is easily compared by computer to a similar string. The mode of operation of MOLGEN-CID is detailed and visualized with examples.

Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t-designs. These are based on assembling systematically orbits of t-homogeneous permutation groups in their induced actions on k-subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3^m − 2, 16·3^m − 3) is obtained. This also provides the smallest known ν for a t-(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001

We show the existence of simple 8-(31,10,93) and 8-(31,10,100) designs. For each value of λ we show 3 designs in full detail. The designs are constructed with a prescribed group of automorphisms PSL(3,5) using the method of Kramer and Mesner KramerMesner76. They are the first 8-designs with small parameters which are known explicitly. We do not yet know if PSL(3,5) is the full group of automorphisms of the given designs. There are altogether 138 designs with λ = 93 and 1658 designs with λ = 100 and PSL(3,5) as a group of automorphisms. We prove that they are all pairwise non-isomorphic. For this purpose, a brief account on the intersection numbers of these designs is given. The proof is done in two different ways. At first, a quite general group theoretic observation shows that there are no isomorphisms. In a second approach we use the block intersection types as invariants, they classify the designs completely.

A halving is a t-design which has the same parameters as its complementary design. Together these two designs form a large set LS[2](t, k, v). There are several recursion theorems for large sets, such that a single new halving results in several new infinite families of halvings. We present new halvings with the parameters 7-(24, 10, 340), 6-(22, 9, 280), 5-(21, 10, 2184), and 5-(21, 9, 910). Recursive constructions by S. Ajoodani-Namini and G. B. Khosrovshahi [Discrete Math 135 (1994), 29–37; J. Combin. Theory A 76 (1996), 139–144] then yield that an LS[2](t, k, v) exists if and only if the parameter set is admissible for t = 6, k = 7, 8, 9, and for t ≤ 5, k ≤ 15. Thus, Hartman's conjecture is true in these cases. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 233–241, 1999

We describe a computer search for the construction of simple designs with prescribed automorphism groups. Using our program package DISCRETA this search yields designs with parameter sets 7-(33, 8, 10), 7-(27, 9, 60), 7-(26, 9, λ) for λ = 54, 63, 81, 7-(26, 8, 6), 7-(25, 9, λ) for λ = 45, 54, 72, 7-(24, 9, λ) for λ = 40, 48, 64, 7-(24, 8, λ) for λ = 4, 5, 6, 7, 8, 6-(25, 8, λ) for λ = 36, 45, 54, 63, 72, 81, 6-(24, 8, λ) for λ = 36, 45, 54, 63, 72, 5-(19, 6, 4), and 5-(19, 6, 6). In several of these cases we are able to determine the exact number of isomorphism types of designs with that prescribed automorphism group. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 79–94, 1999

Up to now, all known Steiner 5-designs are on q+ 1 points where q≡ 3 (mod 4) is a prime power and the design is admitting PSL(2, q) as a group of automorphisms. In this article we present a 5-(36,6,1) design admitting PGL(2, 17) × C2as a group of automorphisms. The design is unique with this automorphism group and even for the commutator group PSL(2, 17) × Id2of this automorphism group there exists no further design with these parameters. We present the incidence matrix of t-orbits and block orbits.

In the present paper we describe the use of group actions, double cosets and homomorphisms in the constructive theory of discrete structures, as we found it useful from both a theoretical and a practical point of view. By means of examples we should like to demonstrate that these methods are useful both as unifying principles and as efficient methods for applications.

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