Description
Springer Classification Algorithms For Codes And Designs 2006 Edition by Petteri Kaski Patric R.J. OEstergard
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) [129] 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 [180] 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. Table of contents : 1 Introduction.- 2 Graphs Designs and Codes.- 3 Representations and Isomorphism.- 4 Isomorph-Free Exhaustive Generation.- 5 Auxiliary Algorithms.- 6 Classification of Designs.- 7 Classification of Codes.- 8 Classification of Related Structures.- 9 Prescribing Automorphism Groups.- 10 Validity of Computational Results.- 11 Computational Complexity.- 12 Nonexistence of Projective Planes of Order 10.- References.- Problem Index.- Index