Designs 2002: Further Computational and Constructive Design Theory
Springer Science & Business Media, Sep 30, 2003 - Mathematics - 368 pages
This volume is a sequel to our 1996 compilation, Computational and Constructive Design Theory. Again we concentrate on two closely re lated aspects of the study of combinatorial designs: design construction and computer-aided study of designs. There are at least three classes of constructive problems in design theory. The first type of problem is the construction of a specific design. This might arise because that one particular case is an exception to a general rule, the last remaining case of a problem, or the smallest unknown case. A good example is the proof that there is no projective plane of parameter 10. In that case the computations involved were not different in kind from those which have been done by human brains without electronic assistance; they were merely longer. Computers have also been useful in the study of combinatorial spec trum problems: if a class of design has certain parameters, what is the set of values that the parameters can realize? In many cases, there is a recursive construction, so that the existence of a small number of "starter" designs leads to the construction of infinite classes of designs, and computers have proven very useful in finding "starter" designs.
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2-HSOLSSOMs AG(d algorithm amicable sets automorphism group balanced ternary base blocks BIBD block design block size candidate arrays chromatic index Combinatorial Designs complete contains corresponding cosets critical set denote disjoint elements entry matrices Example exists Frame of type GDD of type given graph Hadamard matrices Hamming distance distribution hole hyperplane of H i i i i incidence matrix infinite point intersection isomorphism Kirkman Frame Kirkman Triple Systems Koukouvinos large sets latin trade Lemma matrices of order maximal set minimal defining set multiple non-isomorphic NPAF obtain occurs orthogonal design orthogonal STS pairs parameters partial latin square partition permutation prime power problem projections Proof proportionally balanced design resolvable satisfying Seberry Section sequences simulated annealing solutions square of order Steiner latin square Steiner triple systems subgroup subsets Suppose Sylow subgroup symmetric Hamming distance Theorem trade of volume trade volumes triangle-factors wsub