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Finding Simple tDesigns with Enumeration Techniques
 J. Combinatorial Designs
, 1998
"... 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 vset (i.e. a set with v elements) whose elements are called points. A t(v; k; ) design is a ..."
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Cited by 12 (7 self)
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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 vset (i.e. a set with v elements) whose elements are called points. A t(v; k; ) design is a collection of ksubsets (called blocks) of X with the property that any tsubset of X is contained in exactly blocks. A t(v; k; ) design is called simple if no blocks are repeated, and trivial if every ksubset 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 tsubsets of X and the columns by the ksubsets of X. We set m i;j := 1 if the ith tsubset is contained in the jth ksubset, otherwise m i;j := 0. Simple t(v; k; ) designs therefore correspond to ...
The Discovery of Simple 7Designs with Automorphism Group ...
, 1995
"... 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 ..."
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Cited by 10 (7 self)
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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.
Efficient Parallel Solution of Sparse Systems of Linear Diophantine Equations
, 1997
"... 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 smal ..."
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Cited by 10 (3 self)
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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 BlockWiedemann algorithm to solve these preconditioned systems efficiently in parallel. Solutions produced are small and space required is essentially linear in the output size.
Construction of large constant dimension codes with a prescribed minimum distance,” July 2008, available at http://arxiv.org/abs/0807.3212
"... 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 fi ..."
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Cited by 9 (1 self)
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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, qanalogue of Steiner systems, subspace codes 1
Large Sets of Disjoint tDesigns
 Austral. J. Combin
, 1990
"... [u this paper, we show how the basis reduction algorithm of Kreher and Radziszowski 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 rise to an infinite family of large sets of 4desiglls via a result of Tei ..."
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Cited by 5 (0 self)
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[u this paper, we show how the basis reduction algorithm of Kreher and Radziszowski 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 rise to an infinite family of large sets of 4desiglls via a result of Teirlinck [6]. 1
Some Simple 7Designs
 Combinatorial Designs and Related Structures, Proceedings of the First Pythagorean Conference, volume 245 of London Mathematical Society Lecture Notes
"... Some simple 7designs 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 ..."
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Some simple 7designs 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.
Computing Transverse TDesigns
"... In this paper, we develop a computational method for constructing transverse tdesigns. An algorithm is presented that computes the Gorbits of kelement subsets transverse to a partition given that an automorphism group G is provided. We then use this method to investigate transverse Steiner quad ..."
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In this paper, we develop a computational method for constructing transverse tdesigns. An algorithm is presented that computes the Gorbits of kelement 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 tdesigns with t > 3 are also presented. 1
Construction of Combinatorial Objects
, 1995
"... 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, u ..."
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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.
Simple 8Designs with Small Parameters
"... . We show the existence of simple 8(31,10,93) and 8(31,10,100) designs. For each value of 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 [8]. They are the first 8designs with small parameters w ..."
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. We show the existence of simple 8(31,10,93) and 8(31,10,100) designs. For each value of 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 [8]. They are the first 8designs 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 = 93 and 1658 designs with = 100 and PSL(3; 5) as a group of automorphisms. We prove that they are all pairwise nonisomorphic. 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. Keywords: tdesign, KramerMesner method, intersection number, isomorphism problem,...