Lattice basis reduction in combination with an efficient backtracking algorithm is used to find all (4 996 426) simple 7-(33,8,10) designs with automorphism group P\GammaL(2,32). 1 Introduction Let X be a v-set (i.e. a set with v elements) whose elements are called points. A t-(v; k; ) design is a collection of k-subsets (called blocks) of X with the property that any t-subset of X is contained in exactly blocks. A t-(v; k; ) design is called simple if no blocks are repeated, and trivial if every k-subset of X is a block and occurs the same number of times in the design. A straightforward approach to the construction of t-(v; k; ) designs is to consider the matrix M v t;k := (m i;j ); i = 1; : : : ; ` v t ' ; j = 1; : : : ; ` v k ' : The rows of M v t;k are indexed by the t-subsets of X and the columns by the k-subsets of X. We set m i;j := 1 if the i-th t-subset is contained in the j-th k-subset, otherwise m i;j := 0. Simple t-(v; k; ) designs therefore correspond to ...

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.

. Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions. 1. A General Point of View A natural goal in mathematical theories is a full description of the objects that are investigated. This goal has been successfully achieved in some cases, for example all finite abelian groups and with much more effort all finite simple groups. More often one restricted the research activity firstly to more modest problems like the pure existence of any object with som...