Classification Algorithms for Codes and Designs
Springer Science & Business Media, Feb 3, 2006 - Mathematics - 412 pages
A new starting-point and a new method are requisite, to insure a complete [classi?cation of the Steiner triple systems of order 15]. This method was furnished, and its tedious and di?cult execution und- taken, by Mr. Cole. F. N. Cole, L. D. Cummings, and H. S. White (1917)  The history of classifying combinatorial objects is as old as the history of the objects themselves. In the mid-19th century, Kirkman, Steiner, and others became the fathers of modern combinatorics, and their work – on various objects, including (what became later known as) Steiner triple systems – led to several classi?cation results. Almost a century earlier, in 1782, Euler  published some results on classifying small Latin squares, but for the ?rst few steps in this direction one should actually go at least as far back as ancient Greece and the proof that there are exactly ?ve Platonic solids. One of the most remarkable achievements in the early, pre-computer era is the classi?cation of the Steiner triple systems of order 15, quoted above. An onerous task that, today, no sensible person would attempt by hand calcu- tion. Because, with the exception of occasional parameters for which com- natorial arguments are e?ective (often to prove nonexistence or uniqueness), classi?cation in general is about algorithms and computation.
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Representations and Isomorphism 47
IsomorphFree Exhaustive Generation
Classification of Designs 175
Classification of Codes
Classification of Related Structures
Prescribing Automorphism Groups
Validity of Computational Results 297
Nonexistence of Projective Planes of Order 10 339
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1-factorization algorithm approach Aut(X automorphism group backtrack search BIBDs binary blocks canonical augmentation canonical labeling classification algorithms classification problems classification results cliques codewords colored graph columns combinatorial objects complete considered constant weight codes constructed coordinate corresponding cosets covering codes defined Definition designs discussed entries equivalence error-correcting codes Exact Covers Example exists finite functor given graph G graph isomorphism group actions Hadamard matrices homomorphism incidence matrix incidence structure integer intersect invariant isomorph rejection isomorphism classes isomorphism computations Latin squares Lemma Let G lexicographic order linear codes minimum distance morphism node nonisomorphic obtained OE codes orbit transversal ordered partition orthogonal arrays pair parallel classes parameters phism plane of order points polynomial polynomial-time projective plane Proof properties pruning Research Problem rows search tree Sect seeds set system Steiner triple systems subgroup subobject subsets subtree Theorem transformation unique vector vertices