We present a new iterative algorithm for solving large sparse systems of linear Diophantine equations which is fast, provably exploits sparsity, and allows an efficient parallel implementation. This is accomplished by reducing the problem of finding an integer solution to that of finding a very small number of rational solutions of random Toeplitz preconditionings of the original system. We then employ the Block-Wiedemann algorithm to solve these preconditioned systems efficiently in parallel. Solutions produced are small and space required is essentially linear in the output size.

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.

Abstract In this paper we construct constant dimension space codes with prescribed minimum distance. There is an increased interest in space codes since a paper [13] by Kötter and Kschischang were they gave an application in network coding. There is also a connection to the theory of designs over finite fields. We will modify a method of Braun, Kerber and Laue [7] which they used for the construction of designs over finite fields to do the construction of space codes. Using this approach we found many new constant dimension spaces codes with a larger number of codewords than previously known codes. We will finally give a table of the best found constant dimension space codes. network coding, q-analogue of Steiner systems, subspace codes 1

In this paper, we show how the basis reduction algorithm of Kreher and Radziszowski [4] can be used to construct large sets of disjoint designs with specified automorphisms. In particular, we construct a (3,4,23;4)-large set which gives rise to an infinite family of large sets of 4-designs via a result of Teirlinck [6]. 1 Introduction Let X be a finite set of v elements called points. We denote by \Gamma X k \Delta the set of all k-element subsets of X . A t-design, or more specifically, a t-(v; k; ) design, is a pair (X ; B) such that B ` \Gamma X k \Delta , and every member of \Gamma X t \Delta is contained in precisely members of B. The members of B are called blocks. The divisibility conditions \Gamma v\Gammai t\Gammai \Delta j 0 (mod \Gamma k\Gammai t\Gammai \Delta ) for 0 i ! t, provide necessary conditions for the existence of a t-(v; k; ) design. For any given t, k, and v, we denote by (t; k; v) the minimum positive that satisfies the divisibility ...

Some simple 7-designs with small parameters are constructed with the aid of a computer. The smallest parameter set found is 7-(24; 8; 4): An automorphism group is prescribed for finding the designs and used for determining the isomorphism types. Further designs are derived from these designs by known construction processes.

In this paper, we develop a computational method for constructing transverse t-designs. An algorithm is presented that computes the G-orbits of k-element subsets transverse to a partition given that an automorphism group G is provided. We then use this method to investigate transverse Steiner quadruple systems. We also develop recursive constructions for transverse Steiner quadruple systems, and we provide a table of existence results for these designs when the number of points v 24. Finally, some results on transverse t-designs with t > 3 are also presented. 1

Galois rings

. 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...

. Up to now, all known Steiner 5-designs are on q + 1 points where q j 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) \Theta C 2 as a group of automorphisms. The design is unique with this automorphism group and even for the commutator group PSL(2;17) \Theta Id 2 of this automorphism group there exists no further design with these parameters. We present the incidence matrix of t-orbits and block orbits. Keywords: t-design, Steiner system, Kramer-Mesner method. 1. Introduction For a long time, the only known t-designs had t 5 admitting some group PSL(2; q) as a group of automorphisms. The full automorphism group could be larger as in the case of the famous Witt designs [21]. Assmus and Mattson [2] contributed such designs for the cases q = 23; 48 deriving them from codes. The new designs had values of greater than 1, and in several cases consisted of just one orbit of ...