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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.
Isomorphfree exhaustive generation of combinatorial designs
, 2002
"... This report investigates algorithmic isomorphfree exhaustive generation of balanced incomplete block designs (BIBDs), resolvable balanced incomplete block designs (RBIBDs), and the corresponding resolutions. In particular, three algorithms for isomorphfree exhaustive generation of (v, k, λ)BIBDs ..."
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Cited by 3 (0 self)
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This report investigates algorithmic isomorphfree exhaustive generation of balanced incomplete block designs (BIBDs), resolvable balanced incomplete block designs (RBIBDs), and the corresponding resolutions. In particular, three algorithms for isomorphfree exhaustive generation of (v, k, λ)BIBDs and resolutions are described and applied to settle existence and classification problems. The first algorithm generates BIBDs point by point using an orderly algorithm in combination with a maximum clique algorithm. The second and third algorithm generate resolutions of BIBDs by utilizing a correspondence between resolutions and certain optimal qary errorcorrecting codes. The second algorithm generates the corresponding codes codeword by codeword, and is analogous in structure to the first algorithm. The third algorithm generates codes coordinate by coordinate, and is based on the recent isomorphfree exhaustive generation framework of Brendan McKay. The main result of this report is a proof of nonexistence for a (15, 5, 4)
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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Cited by 1 (1 self)
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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.
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,...
A backtracking algorithm for finding tdesigns ∗ M. MNoori a,b B. TayfehRezaie a,1
"... A detailed description of an improved version of backtracking algorithms for finding tdesigns proposed by G. B. Khosrovshahi and the authors of this paper [J. Combin. Des. 10 (2002), 180194] is presented. The algorithm is then used to determine all 5(14,6,3) designs admitting an automorphism of o ..."
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A detailed description of an improved version of backtracking algorithms for finding tdesigns proposed by G. B. Khosrovshahi and the authors of this paper [J. Combin. Des. 10 (2002), 180194] is presented. The algorithm is then used to determine all 5(14,6,3) designs admitting an automorphism of order 13, 11 or 7. It is concluded that a 5(14,6,3) design with an automorphism of prime order p exists if and only if p = 2, 3, 7, 13. 1.
A computer approach to the enumeration of block designs which are invariant with respect to a prescribed permutation group
, 1997
"... ..."
The Simple 7(33,8,10)Designs with Automorphism Group
"... 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 PΓL(2,32). The paper contains a short description of the algorithm. 1 ..."
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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 PΓL(2,32). The paper contains a short description of the algorithm. 1