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Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems.
 Math. Programming
, 1993
"... We report on improved practical algorithms for lattice basis reduction. We propose a practical floating point version of the L3algorithm of Lenstra, Lenstra, Lov'asz (1982). We present a variant of the L3 algorithm with "deep insertions" and a practical algorithm for block KorkinZ ..."
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Cited by 319 (7 self)
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We report on improved practical algorithms for lattice basis reduction. We propose a practical floating point version of the L3algorithm of Lenstra, Lenstra, Lov'asz (1982). We present a variant of the L3 algorithm with "deep insertions" and a practical algorithm for block KorkinZolotarev reduction, a concept introduced by Schnorr (1987). Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 66 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC 1+ computer.
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 15 (8 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 12 (9 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.
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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Cited by 1 (1 self)
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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.