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Topics on Steiner Systems
Handbook of Combinatorics
Handbook of Combinatorics, Volume 1 focuses on basic methods, paradigms, results, issues, and trends across the broad spectrum of combinatorics. The selection first elaborates on the basic graph theory, connectivity and network flows, and matchings and extensions. Discussions focus on stable sets and claw free graphs, nonbipartite matching, multicommodity flows and disjoint paths, minimum cost circulations and flows, special proof techniques for paths and circuits, and Hamilton paths and circuits in digraphs. The manuscript then examines coloring, stable sets, and perfect graphs and embeddings and minors. The book takes a look at random graphs, hypergraphs, partially ordered sets, and matroids. Topics include geometric lattices, structural properties, linear extensions and correlation, dimension and posets of bounded degree, hypergraphs and set systems, stability, transversals, and matchings, and phase transition. The manuscript also reviews the combinatorial number theory, point lattices, convex polytopes and related complexes, and extremal problems in combinatorial geometry. The selection is a valuable reference for researchers interested in combinatorics.
Among the simplest combinatorial designs, triple systems have diverse applications in coding theory, cryptography, computer science, and statistics. This book provides a systematic and comprehensive treatment of this rich area of mathematics.
Combinatorics is a subject of increasing importance, owing to its links with computer science, statistics and algebra. This is a textbook aimed at second-year undergraduates to beginning graduates. It stresses common techniques (such as generating functions and recursive construction) which underlie the great variety of subject matter and also stresses the fact that a constructive or algorithmic proof is more valuable than an existence proof. The book is divided into two parts, the second at a higher level and with a wider range than the first. Historical notes are included which give a wider perspective on the subject. More advanced topics are given as projects and there are a number of exercises, some with solutions given.
Combinatorics is a subject of increasing importance because of its links with computer science, statistics, and algebra. This textbook stresses common techniques (such as generating functions and recursive construction) that underlie the great variety of subject matter, and the fact that a constructive or algorithmic proof is more valuable than an existence proof. The author emphasizes techniques as well as topics and includes many algorithms described in simple terms. The text should provide essential background for students in all parts of discrete mathematics.
The scope of the volume includes all algorithmic and computational aspects of research on combinatorial designs. Algorithmic aspects include generation, isomorphism and analysis techniques - both heuristic methods used in practice, and the computational complexity of these operations. The scope within design theory includes all aspects of block designs, Latin squares and their variants, pairwise balanced designs and projective planes and related geometries.
The study of matroids is a branch of discrete mathematics with basic links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical engineering and statics. This incisive survey of matroid theory falls into two parts: the first part provides a comprehensive introduction to the basics of matroid theory while the second treats more advanced topics. The book contains over five hundred exercises and includes, for the first time in one place, short proofs for most of the subjects' major theorems. The final chapter lists sixty unsolved problems and details progress towards their solutions.
Proceedings of the Conference on Near-Rings and Near-Fields, Stellenbosch, South Africa, July 9-16, 1997
In this monograph, we develop the theory of one of the most fascinating topics in coding theory, namely, perfect codes and related structures. Perfect codes are considered to be the most beautiful structure in coding theory, at least from the mathematical side. These codes are the largest ones with their given parameters. The book develops the theory of these codes in various metrics — Hamming, Johnson, Lee, Grassmann, as well as in other spaces and metrics. It also covers other related structures such as diameter perfect codes, quasi-perfect codes, mixed codes, tilings, combinatorial designs, and more. The goal is to give the aspects of all these codes, to derive bounds on their sizes, and present various constructions for these codes.The intention is to offer a different perspective for the area of perfect codes. For example, in many chapters there is a section devoted to diameter perfect codes. In these codes, anticodes are used instead of balls and these anticodes are related to intersecting families, an area that is part of extremal combinatorics. This is one example that shows how we direct our exposition in this book to both researchers in coding theory and mathematicians interested in combinatorics and extremal combinatorics. New perspectives for MDS codes, different from the classic ones, which lead to new directions of research on these codes are another example of how this book may appeal to both researchers in coding theory and mathematicians.The book can also be used as a textbook, either on basic course in combinatorial coding theory, or as an advance course in combinatorial coding theory.