Finite geometries

**__Finite Geometries__**The authors examine how finite geometries' applicable nature led to solutions of open problems in different fields, such as design theory, cryptography and extremal combinatorics. Other areas covered include proof techniques using polynomials in case of Desarguesian planes, and applications in extremal combinatorics, plus, recent material and developments.Features:Includes exercise sets for possible use in a graduate courseDiscusses applications to graph theory and extremal combinatoricsCovers coding theory and cryptographyTranslated and revised text from the Hungarian published version Cover......Page 1 Half Title......Page 2 Title Page......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 8 1. Definition of projective planes, examples......Page 10 2. Basic properties of collineations and the Theorem of Baer......Page 38 3. Coordinatization of projective planes......Page 56 4. Projective spaces of higher dimensions......Page 84 5. Higher dimensional representations......Page 126 6. Arcs, ovals and blocking sets......Page 142 7. (k; n)-arcs and multiple blocking sets......Page 170 8. Algebraic curves and finite geometry......Page 186 9. Arcs, caps, unitals and blocking sets in higher dimensional spaces......Page 208 10. Generalized polygons, Möbius planes......Page 238 11. Hyperovals......Page 264 12. Some applications of finite geometry in combinatorics......Page 284 13. Some applications of finite geometry in coding theory and cryptography......Page 310 Bibliography......Page 330 Index......Page 342 Finite Geometries stands out from recent textbooks about the subject of finite geometries by having a broader scope. The authors thoroughly explain how the subject of finite geometries is a central part of discrete mathematics. The text is suitable for undergraduate and graduate courses. Additionally, it can be used as reference material on recent works. The authors examine how finite geometries’ applicable nature led to solutions of open problems in different fields, such as design theory, cryptography and extremal combinatorics. Other areas covered include proof techniques using polynomials in case of Desarguesian planes, and applications in extremal combinatorics, plus, recent material and developments. Features: Includes exercise sets for possible use in a graduate course Discusses applications to graph theory and extremal combinatorics Covers coding theory and cryptography Translated and revised text from the Hungarian published version Finite Geometries stands out from recent textbooks on the subject of finite geometries by having a broader scope. This textbook explains the recent proof techniques using polynomials in case of Desarguesian planes.