Anton Betten

Counting symmetric configurations v 3

Abstract In this article we give tables of configurations v 3 for v ⩽18 and triangle-free configurations for v ⩽21 together with some statistics about some properties of the structures like transitivity, self-duality or self-polarity.

Error-correcting linear codes : classification by isometry and applications

Linear Codes.- Bounds and Modifications.- Finite Fields.- Cyclic Codes.- Mathematics and Audio Compact Discs.- Enumeration of Isometry Classes.- Solving Systems of Diophantine Linear Equations.- Linear Codes with a Prescribed Minimum Distance.- The General Case.

Simple 8-Designs with Small Parameters

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.

The Discovery of Simple 7-Designs with Automorphism Group PTL (2, 32)

A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly require small C or C++ programs. The basic philosophy behind the system is to transform problems into standard problems of e.g. group theory, graph theory, linear algebra, graphics, or databases and then to use highly specialized routines from that field to tackle the problems. The transformations required often follow the same principles especially in the case of generation and isomorphism testing.

Unitals and codes

A program is outlined for the enumeration of unital 2-(28,4,1) designs that uses tactical decompositions defined by vectors of certain weight in the dual binary code of a design. A class of designs with a spread that covers a codeword of weight 12 is studied in detail. A total of 909 nonisomorphic designs are constructed that include the classical hermitian and Ree unitals, as well as many other of the 145 previously known 2-(28,4,1) designs.

Linear spaces with at most 12 points

The 28,872,973 linear spaces on 12 points are constructed. The parameters of the geometries play an important role. In order to make generation easy, we construct possible parameter sets for geometries first (purely algebraically). Afterwards, the corresponding geometries are tried to construct. We define line types, point types, point cases, and also refined line types. These are the first three steps of a general decomposition according to the parameters which we call TDO. The depth of parameter precalculation can be varied, thereby obtaining a handy tool to react in a flexible way to different grades of difficulty of the problem.

Tactical decompositions and some configurationsv4

The study of configurations or — more generally — finite incidence geometries is best accomplished by taking into account also their automorphism groups. These groups act on the geometry and in particular on points, blocks, flags and even anti-flags. The orbits of these groups lead to tactical decompositions of the incidence matrices of the geometries or of related geometries. We describe the general procedure and use these decompositions to study symmetric configurationsv4 for smallv. Tactical decompositions have also been used to construct all linear spaces on 12 points [2] and all proper linear spaces on 17 points [3].

The proper linear spaces on 17 points

Abstract The proper linear spaces on 17 points are classified. The computation is based on the parameters of the geometries and makes extensive use of tactical decompositions. A specific one, the tactical decomposition by ordering (TDO) which has been invented by Betten and Braun (Combinatorics’90, Ann. Discrete Math., Vol. 52, North-Holland, Amsterdam, 1992, pp. 37–43) is presented in full detail. The TDO may be seen as the final step of parameters of the geometries. In the current article, the authors show how the TDO can be used in order to construct geometries. This new method starts by calculating all possible TDO-schemes which the requested geometries may have. In a second step, all geometries for a fixed TDO-scheme are constructed. This two-step approach is a versatile tool which may be applied to other construction problems, too. The current work may be seen as an extension of A. Betten and D. Betten (J. Combin. Des. 7 (1999) 119–142) where all linear spaces on 12 points were classified.

A Steiner 5-Design on 36 Points

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) × C2 as a group of automorphisms. The design is unique with this automorphism group and even for the commutator group PSL(2, 17) × Id2 of this automorphism group there exists no further design with these parameters. We present the incidence matrix of t-orbits and block orbits.

Graphical t -designs via polynomial Kramer-Mesner matrices

Kramer-Mesner matrices have been used as a powerful tool to construct t-designs. In this paper we construct Kramer-Mesner matrices for fixed values of k and t in which the entries are polynomials in n the number of vertices of the underlying graph. From this we obtain an elementary proof that with a few exceptions Sn[2] is a maximal subgroup of Sn2 or An2. We also show that there are only finitely many graphical incomplete t-(v,k,λ) designs for fixed values of 2 ⩽ t and k at least in the cases k = t + 1, t = 2, and 2 ⩽ t < k ⩽ 6. All graphical t-designs are determined by the program DISCRETA3 for various small parameters. Most parameter sets are new for graphical designs, some also for general simple t-designs. The largest value of t for which graphical designs were found is t = 5. Some of the smaller designs which are block transitive are drawn as graphs.