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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.
Steiner systems S(5,6, v) with . . .
, 1998
"... It is proved that there are precisely 4204 pairwise nonisomorphic Steiner systems S(5, 6, 72) invariant under the group PSL2(71) and which can be constructed using only short orbits. It is further proved that there are precisely 38717 pairwise nonisomorphic Steiner systems S(5, 6, 84) invariant u ..."
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It is proved that there are precisely 4204 pairwise nonisomorphic Steiner systems S(5, 6, 72) invariant under the group PSL2(71) and which can be constructed using only short orbits. It is further proved that there are precisely 38717 pairwise nonisomorphic Steiner systems S(5, 6, 84) invariant under the group PSL2(83) and which can be constructed using only short orbits.
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 Steiner 5Design on 36 Points
"... . Up to now, all known Steiner 5designs 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 t ..."
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. Up to now, all known Steiner 5designs 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 torbits and block orbits. Keywords: tdesign, Steiner system, KramerMesner method. 1. Introduction For a long time, the only known tdesigns 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 ...